.

:

pi

v. 0.86

.

pi 2010

.

iii

v

1 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 132.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 243.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 334.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 395.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6 526.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

i

.

ii

7 677.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8 758.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

9 pi 839.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

10 9110.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9310.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

11 10111.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10411.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

12 Taylor 11112.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11112.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11312.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

123.1 . . . . . . . . . . . . . 123

pipi 126.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

145.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

pi 160.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

.

pi pi , 1 . pi pi pi , pi pi .

pi pi ( pi ) .

1. pi pi.

2. pi.

3. pi pi pi pi.

, pi . , pi pi , pi pi pi . pi-, pi pi , pi . pipi pi pi . , ( . ).

(pi pi pi ) pi pi ... .

: pipi , pi pi pi pi ... pipi pipi . pi , pipi pi pi pipi. , pipi pi pi. pi pi pi pi .

pi pi pi - pi . pi pi pi

1. pi

iii

.

iv

pi pi , pi pi .

pi . pi pi pi pi . pi - ( pi ) . pi v.0.85 ., pi pi- , beta. pi pi pi v.1.00. pi (pi) pi pi .

, pi 2009

.

( Calculus) pi pi , pi 2 pi . pi . , ( ) pi pi : f (x) f = f (x+ x)f (x) x x pipi pipi pi x pi , pi . pi pi .

pi pi - ( ) 3. pi pi 17 . pipi, pi pi pipi pi pi pi pi. pi pi pipi pi pi- (pi.. pi , pi pi ) pi (pi pi ) pi pi.

pi pi . f (x) pi pi pi : f : R R. pi - . pi pi .

pi pi , pi - pi . .

1. 1 i: 1 = i, i2 = 1.2. pi R

C.

3. :N

n=1 an = a1 + a2 + ...+ aN .

2 pi .3.. pi .

v

.

vi

4. .

.

1

ex pi . - lnx .

( sinx), ( cosx), pi (tanx = sinx

cosx), pi (cotx = cosx

sinx),

(secx = 1cosx

) (cscx = 1sinx

). pi . ( ) .

pi pi pi .

1.1

1.1.1. ex .

1.1.2. pi x 1

fn (x) =(

1 +x

n

)(

1 +x

n

) ...

(1 +

x

n

) .

n

x R f1 (x), f2 (x) , ... pi0 f1 (x) f2 (x) ... fn (x) fn+1 (x) ... ex

, n pi, fn (x) ex, pi :(1 +

x

n

)nn

ex.

1.1.3. lnx :

y = lnx x = ey. .

1 11.

1

.

1. 2

1.1.4. pi pi pi pi pi

eix = cosx+ i sinx (1.1)

cos (x) = cos x, sin (x) = sinx. (1.2)

(1.1), (1.2) pi . 12.

1.1.5. pi (1.1), (1.2) pi

cosx =eix + eix

2, sinx =

eix eix2i

. (1.3)

1.1.6.

sinh (ix) = i sinx,

sin (ix) = i sinhx,

cosh (ix) = cos x,

cos (ix) = cosh x.

1.1.7. .

arcsin(x) = y x = sin(y)arccos(x) = y x = cos(y)arccos(x) = y x = tan(y)arccot(x) = y x = cot(y)arcsec(x) = y x = sec(y)arccsc(x) = y x = csc(y).

1.1.8. pi , pi pi pi pi, pi .

pi : sinh(x) = ex ex

2

pi : cosh(x) = ex + ex

2

pi pi : tanh(x) = ex exex + ex

pi pi : coth(x) = ex + ex

ex expi : sech(x) = 2

ex + ex

pi : csch(x) = 2ex ex .

.

1. 3

1.1.9. pi .

arc sinh (x) = y x = sinh(y)arc cosh (x) = y x = cosh(y)arc tanh(x) = y x = tanh(y)arc coth(x) = y x = coth(y)arc sech(x) = y x = sech(y)arc csch(x) = y x = csch(y).

1.1.10.

arc sinh (x) = ln(x+x2 + 1

), < x 1 x < 1 (1.7)

arc sech(x) = ln

(1

x+

1 x2x

), 0 < x 1 (1.8)

arc csch(x) = ln

(1

x+

1 + x2

|x|

), x 6= 0. (1.9)

1.2

1.2.1. pi (1.3) (1.1) x x

eix = cos (x) + i sin (x) = cos x i sinx (1.10)(pi pi (1.2) ). (1.1) (1.10) pi

cosx =eix + eix

2.

pi

sinx =eix eix

2i.

1.2.2. pi cos2 x+ sin2 x = 1.

cos2 x+ sin2 x =

(eix + eix

2

)2+

(eix + eix

2i

)2=ei2x + ei2x + 2

4 e

i2x + ei2x 24

=4

4= 1.

.

1. 4

1.2.3. pi sin(a+ b) = sin a cos b+ cos a sin b

sin a cos b+ cos a sin b =eia eia

2i e

ib + eib

2+eia + eia

2 e

ib eib2i

=ei(a+b) ei(a+b) + ei(ab) ei(a+b)

4i

+ei(a+b) + ei(a+b) ei(ab) ei(a+b)

4i

=2ei(a+b) 2ei(a+b)

4i= sin (a+ b) .

1.2.4. pi sin a = tan a1+tan2 a

tan a1 + tan2 a

=sin a/ cos a

1 + sin2 a/ cos2 a=

sin a/ cos a(cos2 a+ sin2 a

)/ cos a

= sin a.

1.2.5. pi sin(2a) = 2 sin a cos a

2 sin a cos a = 2 eia eia

2i e

ia + eia

2

=eia+ia eiaeia + eiaia eaia

2i

=ei2a ei2a

2i= sin (2a) .

1.2.6. pi sin a cos b = sin(ab)+sin(a+b)2

sin(a b) + sin(a+ b)2

=sin a cos b cos a sin b+ (sin a cos b+ cos a sin b)

2= sin a cos b.

1.2.7. pi sin a+ sin b = 2 sin(a+b

2

)cos(ab

2

) pi 1.2.6

sinA cosB =sin(AB) + sin(A+B)

2.

A = a+b2

, B = ab2

, AB = b, A+B = a, pi

sin

(a+ b

2

)cos

(a b

2

)=

sin(a) + sin(b)

2

pi pi pipi .

.

1. 5

1.1: pi coshx.

1.2.8. coshx. coshx = ex+ex

2. 1.1.

pi ex, ex. pi coshx, pi . limx coshx = .pi pi coshx cosh 0 = 1. pi pi pi coshx ( !) , pi pi, :

coshx 1 = ex + ex

2 1 = e

x + ex 22

=

(ex/2 ex/2)2

2 0

coshx 1 = 0 ( coshx = 1) pi

ex/2 ex/2 = 0 ex/2 = ex/2 ex = 1 x = 0.

1.2.9. sinhx. sinhx = exex

2. 1.2.

pi ex, ex. pi sinhx, pi . limx sinhx = limx sinhx =. pi

0 = sinhx =ex ex

2 ex = ex x = 0

. sinhx = 0 x = 0.

1.2.10. tanhx.

.

1. 6

1.2: pi sinhx.

tanhx =sinhx

coshx=ex exex + ex

.

pi

limx

tanhx = limx

tanhx = limx

ex exex + ex

= limx

1 e2x1 + e2x

=1 01 + 0

= 1.

pi limx tanhx = 1 pi tanhx = 0 x = 0. pi pi 1.3.

1.2.11. pi

sinh (x) = sinhx, cosh (x) = coshx. (1.11)

sinh (x) = ex e(x)

2= e

x ex2

= sinhx.

cosh (x) = cosh x pi pi.1.2.12. pi cosh2 x sinh2 x = 1.

cosh2 x sinh2 x =(ex + ex

2

)2(ex ex

2

)2=e2x + e2x + 2

4 e

2x + e2x 24

=4

4= 1.

.

1. 7

1.3: pi tanhx.

1.2.13. pi sinh(x+ y) = sinh x cosh y + coshx sinh y

sinhx cosh y + coshx sinh y =ex ex

2 e

y + ey

2+ex + ex

2 e

y ey2

=ex+y ex+y + exy e(x+y)

4

+ex+y + ex+y exy e(x+y)

4

=2ex+y 2e(x+y)

4= sinh (x+ y) .

1.2.14. pi sinh(2x) = 2 sinhx coshx pi 1.2.13 y = x

sinh (2x) = sinh(x+ x) = sinh x coshx+ coshx sinhx = 2 sinh x coshx.

1.2.15. pi sinhx+ sinh y = 2 sinh(x+y

2

)cosh

(xy

2

)

2 sinh

(x+ y

2

)cosh

(x y

2

)= 2 e

x+y2 ex+y2

2 e

xy2 + e

xy2

2

=ex+y+xy

2 exy+xy2 + ex+yx+y2 exyx+y22

=ex ex

2+ey ey

2= sinhx+ sinh y.

.

1. 8

1.2.16. pi arc sinh(x) = ln

(x+x2 + 1

) z = arc sinh(x).

x = sinh z =ez ez

2 ez 2x ez = 0.

a = ez, pi a 2x a1 = 0 pipi a, pipi

a2 2xa 1 = 0. pi a pi

ez = a = xx2 + 1.

a = xx2 + 1 pipi ( a = ez > 0). pi

a = ez = x+x2 + 1 z = ln

(x+x2 + 1

).

1.2.17. pi arc sinh (x) R. arc sinh (x) = ln

(x+x2 + 1

). pi

pi x2 + 1 > |x| , pi x +x2 + 1 > 0

ln(x+x2 + 1

)pi pipi x R (

pipi ). pi 1.4 pi arc sinh (x), pi sinhx pi x = y ( ;). pi pi x R pi arc sinhx.

.

1. 9

1.4: pi arc sinhx.

1.2.18. pi arc cosh (x) [1,). arc cosh (x) = ln

(x+x2 1). pi, pi x

pi pi

x2 1 0 x+x2 1 > 0.

pi pi , pipi |x| 1. pi |x| > x2 1, pi pipi x > 0. pi x > 0 |x| 1pipi x 1, . , pi pi pi pi pi arc coshx, pi 1.5.

1.5: pi arc coshx.

1.2.19. arc tanh(x) = 12

ln(

1+x1x), 1 < x < 1.

pi pi pi arc tanhx, pi 1.6 (pi tanhx pi x = y).

.

1. 10

1.6: pi arc tanhx.

pi pi pi pi x 1+x1x > 0 ( !).

1.3

1.3.1. pi pi pi.

1. sin(a b) = sin a cos b cos a sin b2. cos(a b) = cos a cos b sin a sin b3. tan(a b) = tan atan b

1tan a tan b

1.3.2. pi cos a = 11+tan2 a

.

1.3.3. pi pi pi.

1. cos(2a) = cos2 a sin2 a = 2 cos2 a 12. tan(2a) = 2 tan a

1tan2 a .

1.3.4. pi pi pi.

1. sin(3a) = 3 sin a 4 sin3 a2. cos(3a) = 4 cos3 a 3 cos a3. tan(3a) = 3 tan atan3 a

13 tan2 a .

1.3.5. pi pi pi.

1. sin a sin b = cos(ab)cos(a+b)2

.

1. 11

2. cos a cos b = cos(ab)+cos(a+b)2

1.3.6. pi pi pi.

1. sin a+ sin b = 2 sin(a+b

2

)cos(ab

2

)2. cos a+ cos b = 2 cos

(a+b

2

)cos(ab

2

)3. cos a cos b = 2 sin (a+b

2

)sin(ba

2

)1.3.7. pi 1 tanh2(x) = 1

cosh2(x).

1.3.8. pi

1. sinh(x y) = sinh x cosh y coshx sinh y2. cosh(x y) = cosh x cosh y sinhx sinh y3. tanh(x y) = tanhxtanh y

1tanhx tanh y

1.3.9. pi

1. cosh(2x) = cosh2 x+ sinh2 x = 2 cosh2 x 12. tanh(2x) = 2 tanhx

1+tanh2 x.

1.3.10. pi

1. sinhx sinh y = 2 cosh (x+y2

)sinh

(xy

2

)2. coshx+ cosh y = 2 cosh

(x+y

2

)cosh

(xy

2

)3. coshx cosh y = 2 sinh (x+y

2

)sinh

(xy

2

).

1.3.11. pi

1. sinh (ix) = i sinx

2. sin (ix) = i sinhx

3. cosh (ix) = i cosx

4. cos (ix) = i coshx

1.3.12. pi

1. arc sinh(x) = ln(x+x2 + 1

)( < x

.

1. 12

4. arc coth(x) = 12

ln(x+1x1)(x > 1 x < 1),

5. arc sech(x) = ln(

1x

+

1x2x

), 0 < x 1

6. arc csch(x) = ln(

1x

+

1+x2

|x|

), x 6= 0.

1.3.13. pi

1. arcsin(x) = i ln (ix+1 x2),2. arccos(x) = i ln (x+x2 1),3. arctan(x) = i

2ln(

1ix1+ix

).

.

2

pi pi , pi .

2.1

2.1.1. f (x) f0 ( f0) x x0 :

> 0 : > 0 : 0 < |x x0| < |f (x) f0| < . (2.1)

limxx0

f (x) = f0.

2.1.2. (2.1) : f (x) f0 ( ), x pi x0 ( 0 < |x x0| < ). (2.1) pipi x = x0 (0 < |x x0|).2.1.3. f (x) ( ) x x0 :

M > 0 : > 0 : 0 < |x x0| < f (x) > M. (2.2)

f (x) ( ) x x0 :

M < 0 : > 0 : 0 < |x x0| < f (x) < M. (2.3)

limxx0

f (x) =, limxx0

f (x) = .

13

.

2. 14

2.1.4. f (x) f0 ( f0) x :

> 0 : M > 0 : M < x |f (x) f0| < . (2.4)

f (x) f0 ( f0) x :

> 0 : M < 0 : x < M |f (x) f0| < . (2.5)

limx

f (x) = f0, limx

f (x) = f0.

2.1.5. pi

limx

f (x) =, limx

f (x) = , limx

f (x) =, limx

f (x) = .

2.1.6. f (x) f0 x x0 :

> 0 : > 0 : 0 < x0 x < |f (x) f0| < . (2.6)

f (x) f0 x x0 :

> 0 : > 0 : 0 < x x0 < |f (x) f0| < . (2.7)

limxx0

f (x) = f0, limxx+0

f (x) = f0.

pi .

2.1.7. pi

limxx0

f (x) =, limxx+0

f (x) =, limxx0

f (x) = , limxx+0

f (x) = .

2.1.8. f (x) pi limxx0 f (x)

limxx0

f (x) = limxx0

f (x) = limxx+0

f (x) .

, pi pi limxx0 f (x) = limxx+0 f (x) pi limxx0 f (x) pi.

2.1.9. pi pipi , -pi pi

n R : limxx0

xn = xn0 ( xn0 )

.

2. 15

limxx0 f (x) = f0limxx0 g (x) = g0

}

limxx0 (kf (x)) = kf0limxx0 (f (x) g (x)) = f0 g0limxx0 (f (x) g (x)) = f0 g0limxx0

(f(x)g(x)

)= f0

g0( g0 6= 0)

limxx0 (f (x))n = fn0 ( fn0 )

2.1.10. limxx0 f (x) = f0 limxf0 g (x) = g0 , limxx0 (g (f (x))) = g0.

2.1.11. limxx0 f (x) = f0, limxx0 g (x) = g0, limxx0 h (x) = h0 f (x) g (x) h (x), f0 g0 h0.2.1.12. limxx0 f (x) = limxx0 g (x) = , limxx0 (f (x) + g (x)) =,limxx0 (f (x) g (x)) =.2.1.13. limxx0 f (x) = limxx0 g (x) = , limxx0 (f (x) + g (x)) =, limxx0 (f (x) g (x)) =.2.1.14. f (x) x0

limxx0

f (x) = f (x0) . (2.8)

pipi pi limxx0 f (x) f (x0).

2.1.15. (2.8) , f (x) x0. pi pi pi limxx0 f (x), pi f (x0), pilimxx0 f (x) 6= f (x0).2.1.16. f (x) A R x0 A. pi f (x) ( pi A), f (x) pi .

2.1.17. f (x), g (x) x0, kf (x), f (x) g (x), f (x) g (x),

(f(x)g(x)

)( f (x0) 6= 0).

2.1.18. pi .

2.1.19. ex, sin (x), cos (x) . pi .

2.1.20. [a, b] R f (x) [a, b], pi f (a) < f (b). f0 [f (a) , f (b)] pi x0 [a, b] f0 = f (x0).

.

2. 16

2.2

2.2.1. limx2 (2x 1) = 3.. pi

> 0 : > 0 : 0 < |x 2| < |2x 1 3| < . pi > 0. = 2 x |x 2| < = 2

|2x 1 3| = |2x 4| = 2 |x 2| < 2 = 22

=

pi .

2.2.2. limx0 (x2 + 1) = 1.. pi

> 0 : > 0 : 0 < |x 0| < x2 + 1 1 < .

pi > 0. = pi x |x 0| =

|x| < = x2 + 1 1 = x2 = |x|2 < 2 = ()2 =

pi .

2.2.3. limx1 x23x+2(x2)2 = 0.

. pi

> 0 : > 0 : 0 < |x 1| < x2 3x+ 2(x 2)2

< . pi > 0. = 1+ x |x 1| < =

1+ x2 3x+ 2(x 2)2

= (x 1) (x 2)(x 2)2 = x 1x 2

., |x 1| <

|x 2| > ||x 1| 1| = |1 |x 1|| > 1 ( |x 1| < ) pi x 1x 2

< 1 ., ,

1 < .

1 < < (1 ) <

+ < (1 + ) < < 1 +

.

pi.

.

2. 17

2.2.4. (2.2) 2.3);. pi limxx0 f (x) = (2.2) .

M > 0 : M > 0 : 0 < |x x0| < M f (x) > M. , limxx0 f (x) = M ( ) pi M , x x0 (. |x x0| < M ) f (x) M . ., pi , pi f (x) , pi x x0. (2.3) pi.

2.2.5. limx0 1x2 =.. pi

M > 0 : M > 0 : 0 < |x 0| < M 1x2

> M.

pi M > 0. M = 1M pi

|x| = |x 0| < M = 1M x2 < 1

M

1x2 = 1x2 > 11

M

= M

pi .

2.2.6. (2.4) (2.5);. pi limx f (x) = f0 (2.4) .

> 0 : M > 0 : M < x |f (x) f0| < . , limx f (x) = f0 ( ) pi M , x M f (x) f0 ( pi ) . ., pi , pi f (x) f0 , pi x . (2.5) pi.

2.2.7. limx 1x = 0 limx1x

= 0.. pi pipi

< 0 : M > 0 : M < x1x < .

pi > 0. M = 1 x x > M =1> 0

1x = 1x < 1M = 11 =

pi . pi pi.

.

2. 18

2.2.8. limx x2 =.. pi

M > 0 : NM > 0 : NM < x x2 > M. M pi NM =

M .

2.2.9. limx x3 = .. pi

M < 0 : NM < 0 : x < NM x3 < NM . M < 0 pi NM = M1/3.

2.2.10. limx0 1x = .. pi

M < 0 : M > 0 : 0 < 0 x < M x < M. pi M < 0. M = 1M x 0 < 0 x 0 : M > 0 : 0 < x 1 < M x+ 1x2 1 > M.

pi M > 0. M = 1M x 0 < x 1 11M

= M

pi .

2.2.12. , pipi pi pi (x), (x), limxx0 pi (x) =pi (x0) limxx0 (pi (x) + (x)) = pi (x0) + (x0).

. pi (x) = a0 + a1x+ ...+ aNxN .

limxx0

pi (x) = limxx0

(a0 + a1x+ ...+ aNx

N)

= limxx0

(a0) + limxx0

(a1x) + ...+ limxx0

(aNx

N)

= a0 + a1 limxx0

x+ ...+ aN limxx0

xN

= a0 + a1x0 + ...+ aNxN0 = pi (x0) .

.

2. 19

pi pi pi 2.1.9. limxx0 (x) = (x0). pi, 2.1.9

limxx0

(pi (x) (x)) =(

limxx0

pi (x)

)(

limxx0

(x)

)= pi (x0) (x0) .

2.2.13. pi pi

1. limx1 (5x2 + 3x 4).2. limx1 x2x24x+1

3. limx4x4.

4. limx1

5x2 + 3x 4.. (1) , 2.2.12,

limx1

(5x2 + 3x 4) = 5 (1)2 + 3 (1) 4 = 2.

(2)

limx1

x 2x2 4x+ 1 =

limx1 (x 2)limx1 (x2 4x+ 1) =

1 212 4 1 + 1 =

12 =

1

2

pi pi 2.1.9, pi - . (3) , pi pipi 2.1.9 n = 1/4, limx4

x4 =

44 = 16. (4) g (x) =

x, f (x) = 5x2 + 3x 4

, 2.1.10,

limx1

5x2 + 3x 4 =

limx1

(5x2 + 3x 4) =

4 = 2.

2.2.14. , limxx0 f (x) = f0 limxx0 g (x) = g0,

limxx0

(f (x) + g (x)) = f0 + g0.

. pi pi

1 > 0 : 1 > 0 : 0 < |x x0| < 1 |f (x) f0| < 1, (2.9)2 > 0 : 2 > 0 : 0 < |x x0| < 2 |g (x) g0| < 2. (2.10)

. (2.9)-(2.10) pi 1 = 2 = 2 pi pi = min (1, 2). pi

0 < |x x0| < < min (1, 2){ |f (x) f0| < 1 = 2|g (x) g0| < 2 = 2

} |f (x) + g (x) (f0 + g0)| |f (x) f0|+ |g (x) g0| <

2+

2=

.

.

2. 20

2.2.15. pi pi

1. limx1 x1x23x+2 .

2. limx1 x21x41 .

3. limx1 1x2

2x2+3 .

4. limh0 (x+h)2x2h

.

. (1)

limx1

x 1x2 3x+ 2 = limx1

x 1(x 1) (x 2) = limx1

1

x 2 =1

1 = 1.

(2)

limx1

x2 1x4 1 = limx1

(x 1) (x+ 1)(x 1) (x+ 1) (x2 + 1) = limx1

1

x2 + 1=

1

2.

(3)

limx1

1 x22x2 + 3 = limx1

(1 x2

2x2 + 3 2 +x2 + 3

2 +x2 + 3

)

= limx1

((1 x2) 2 +x2 + 3

22 x2 3

)

(4)

limh0

(x+ h)2 x2h

= limh0

(x2 + 2xh+ h2) x2h

= limh0

2xh+ h2

h= lim

h0(2x+ h) = 2x.

2.2.16. pi pi

1. limx x+3x+4 .

2. limx 2x+45x1 .

3. limx x4x21 .

4. limx x24x

.

.

2. 21

. (1)

limx

x+ 3

x+ 4= lim

x

xx

+ 3x

xx

+ 4x

=limx xx + limx

3x

limx xx + limx4x

=1 + 0

1 + 0= 1

pi: limx xx = limx 1 = 1 limx3x

= 3 limx 1x = 3 0 ( pi limx 1x = 0) , pi, limx

4x

= 0., (2)

limx

2x+ 4

5x 1 = limx2xx

+ 4x

5xx 1

x

=2 + 0

5 0 =2

5.

(3)

limx

x 4x2 1 = limx

xx2 4

x2

x2

x2 1

x2

=0 01 0 = 0.

(4)

limx

x2 4x

= limx

(x 4

x

)= 0 =.

2.2.17. f (x) = x2 + 2x x0 = 3.. f (x) pi, pi 2.2.12 ( x0 = 3)

limx3

f (x) = f (x0)

. x0 = 3. x0 R.2.2.18. f (x) = x3+2x

x2+1 R.

. x0 R (pi 2.1.19) limxx0 =f (x0) pi pi pi f (x). x2 +1 pi R, . f (x) pi R.

2.2.19. f (x) = |x| R.. pi

f (x) = |x| ={ x x < 0

x x 0 .

x0 (0,)

limxx0

f (x) = x0 = f (x0)

x0 (, 0)

limxx0

f (x) = x0 = f (x0)

.

2. 22

f (x) R{0}. x0 = 0; pi

limx0

f (x) = 0, limx0+

f (x) = 0

limx0

f (x) = 0 = f (0) .

f (x) = |x| R.2.2.20. f (x) = x2

x24 .. f (x) pi pi .

f (x) pi pi pi-, . 2, 2. , pi .

2.3

2.3.1. limx2 (2x 1) = 3.2.3.2. limx2

x 1 = 1.

2.3.3. limx1 x21x1 = 2.

2.3.4. (2.2) ;;);

2.3.5. limx 1x = 0 limx1x

= 0.

2.3.6. (2.4) (2.5);

2.3.7. limx x2 =.2.3.8. limx x3 = .2.3.9. limx0 1x = .2.3.10. limx1+ x+1x21 =.2.3.11. pi pi

1. limx1 (5x2 + 3x 4).2. limx4

x4.

3. limx1

5x2 + 3x 4.4. limx3 x

25x+6x28x+16 .

pi. 2, 16, 2, 0.2.3.12. , pipi pi pi (x), (x), limxx0 (pi (x) (x)) =pi (x0) (x0).

.

2. 23

2.3.13. , limxx0 f (x) = f0 limxx0 g (x) = g0,

limxx0

(f (x) g (x)) = f (x0) g (x0) .2.3.14. pi pi

1. limx1 x2x25x+6 .

2. limx1 x21x31 .

3. limh0 (x+h)3x3h

.

4. limx2 4x2

3x2+5 .

pi. 1/2, 2/3, 3x2, 2.2.3.15. pi pi

1. limx1 x2x1 .

2. limx1+ x2x1 .

3. limx0+ 17x2 .

4. limx0 3x24x .

pi. ,,,,.2.3.16. pi pi

1. limx x2x+1 .

2. limx 3x47x1 .

3. limx 3x47x21 .

4. limx 3x24

7x1 .

5. limx 3x24

7x1 .

pi. 1, 3/7, 0,,.2.3.17. limx0 sinxx = 1.2.3.18. f (x) = x2 + 2x x0 = 3.2.3.19. f (x) = x3+2x

x2+1 R.

2.3.20. f (x) = |x| R.2.3.21. f (x), g (x) x0, f (x g (x)) pi x0.2.3.22. f (x) = x1

x21 .pi. 1, 1.

.

3

pi f (x) f (x) f (x). pi pi pi. pi .

3.1

3.1.1. pi f (x) f (x)

f (x) = limx0

f (x+ x) f (x)x

.

3.1.2. pi f (x) /f (x) pi.

f (x) f (x)1 0x 1x2 2xxn nxn1

sinx cosxcosx sinxtanx 1

cos2 x

sinhx coshxcoshx sinhxex ex

lnx 1x

3.1.3. pi pi pi. pi F (x) pi pi F (x)

24

.

3. 25

.F (x) F (x)f (x) f (x)f (x) + g (x) f (x) + g (x)c f (x) c f (x) (pi c )f (x) g (x) f (x) g (x) + f (x) g (x)f(x)g(x)

f(x)g(x)+f (x)g(x)g2(x)

( g (x) 6= 0)f (h (x)) f (h (x)) h (x)

3.1.4. pi f (x) df

dx= lim

x0f

x= f (x)

pif = f (x+ x) f (x)

f x x.

3.1.5. pi f (x) ' fx

f ' f (x) x. (3.1) pi |x|.3.1.6. df

dx pi f

pi x x f pi .

3.1.7. dfdx

, pi , pi..

df = f (x) dx. (3.2) pi df (3.2) f (x). (3.1) f pipi f (x) x |x| . pi pi , df = f (x) dx pi (pi.. pi ) pi df

dx ( ).

3.1.8. y (x) pi , pi P (x, y) = 0. x y pi (-) , pi P (x, y) = 0 y (x) x. , pi pi y (x) x y ( pi 3.2.19 3.2.22).

3.2

3.2.1. pi pi f (x) = x .

f (x) = limx0

f (x+ x) f (x)x

= limx0

x+ x xx

= limx0

x

x= lim

x01 = 1.

.

3. 26

3.2.2. pi pi f (x) = x2 .

f (x) = limx0

f (x+ x) f (x)x

= limx0

(x+ x)2 x2x

= limx0

x2 + 2x x+ (x)2 x2x

= limx0

(2x+ x) xx

= limx0

(2x+ x) = 2x.

3.2.3. pi (c f (x)) = c f (x) .

(c f (x)) = limx0

c f (x+ x) c f (x)x

= c limx0

f (x+ x) f (x)x

= c f (x) .

3.2.4. pi pi f (x) = x2 4x+ 5. (

x2 4x+ 5) = (x2) 4 (x) + (5) = 2x 4 + 0 = 2x 4.3.2.5. pi pi f (x) = x24x+5

x2+1.

(x2 4x+ 5x2 + 1

)=

(x2 4x+ 5) (x2 + 1) (x2 4x+ 5) (x2 + 1)(x2 + 1)2

=(2x 4) (x2 + 1) (x2 4x+ 5) 2x

(x2 + 1)2=

4x2 8x 4(x2 + 1)2

.

3.2.6. pi pi f (x) = ex (x2 + 1). (

ex (x2 + 1)) = ex (x2 + 1)+ ex 2x = ex (x2 + 2x+ 1) .3.2.7. pi pi f (x) = ex24x+5.

(ex

24x+5)

= (2x 4) ex24x+5.3.2.8. pi pi f (x) = ln (x2 sinx).

[ln(x2 sinx

)]=

1

x2 sinx (2x sinx+ x2 cosx

).

3.2.9. pi pi y (x) = arcsin x.

y = arcsinx x = sin y,pi pi pi cos y =

1 x2.

dx

dy= cos y dy

dx=

1dxdy

=1

cos y=

11 x2 .

.

3. 27

3.2.10. pi pi y (x) = arctan x.

y = arctanx x = tan y,pi pi pi cos y =

1 + x2.

dx

dy= tan y dy

dx=

1dxdy

=11

cos2 y

=1

1 + x2.

3.2.11. pi (sin(x)) = cos(x).

(sinx) =(eix eix

2i

)=ieix (i) eix

2i=eix + eix

2= cosx.

3.2.12. pi (sinh(x)) = cosh(x).

(sinhx) =(ex ex

2

)=ex + ex

2= coshx.

3.2.13. pi (arcsinx) = 11x2 .

z = arcsinx. x = sin z

dx

dz= cos z =

1 sin2 z =

1 x2

pi(arcsinx) =

dz

dx=

11 x2 .

3.2.14. pi (arc sinh(x)) = 1x2+1

. z = arc sinhx. x = sinh z

dx

dz= cosh z =

1 + sinh2 z =

1 + x2

pi(arc sinhx) =

dz

dx=

11 + x2

.

3.2.15. pi f (x) = x2 .

df = f (x) dx = 2xdx.

. pi x. f (x) = x2 . pi pi x x + x. pi 3.1. x , pi pi pi x x 2xx. pi pipi (x)2 pi pi x, x , (x)2

.

3. 28

x

x x

x

x

x

3.1: .

pi 2xx pi . .., x = 2 x = 0.1,

(x+ x)2 = 2.12 = 4. 41, x2 = 22 = 4,

(x+ x)2 x2 = 4.41 4 = 0.41,2xx = 2 2 0.1 = 0.4,(x)2 = (0.1)2 = 0.01,

. f = 0.41 pipi pi 2xx = 0.4.

3.2.16. pi

4.1 pi . f (x) =

x.

x+ x = f (x+ x) f (x) + f (x) x = x+ 1

2x

x.

pi x = 4, x+ x = 4.1 x = 0.1, pipi

4.1 =

4 + 0.1

4 +1

2

40.1 = 2 +

0.1

4= 2. 025.

pi

4.1 = 2. 024 8.

|2.0248 2.025|2.048

= 9. 765 6 105

pi .

.

3. 29

3.2.17. pi

4.01 pi . pi pi pi , pi

x = 4, x+ x = 4.01 x = 0.01, pi

4.01 =

4 + 0.01

4 +1

2

40.01 = 2 +

0.01

4= 2. 002 5.

pi

4.01 = 2.002498439.

|2.002498439 2.0025|2.002498439

= 7. 795 3 107

. pi pi

4.1. ( pi) x, pi .

3.2.18. pi sin (10) pi . cos (610) cos (440).

x . pipi 10 .

x

pi=

10

1800 x = 1. 745 3 102.

pi

sin(10)

= sin(1. 745 3 102) sin (0) + sin (0) 1. 745 3 102

= sin (0) + cos (0) 1. 745 3 102 1. 745 3 102.

cos(610)

= cos(pi

3+ 1. 745 3 102

) cos (pi/3) sin (pi/3) 1. 745 3 102

=1

2

3

2 1. 745 3 102 0.484 89.

cos(440)

= cos(pi

4+ 1. 745 3 102

) cos (pi/4) sin (pi/4) 1. 745 3 102

=

2

2

2

2 1. 745 3 102 0.694 77.

3.2.19. y y3 + y2 5y x2 + 4 = 0. g (x) = g (y (x)) = (y (x))3 , . g (y) = y3.

dg

dx=dg

dy

dy

dx= 3y2y. (3.3)

,d

dxy2 = 2yy. (3.4)

.

3. 30

pi (3.3) (3.4)

d

dx

(y3 + y2 5y x2 + 4) = d

dx(0)

3y2y + 2yy 5y 2x+ 0 = 0y (3y2 + 2y 5) = 2x

y =2x

3y2 + 2y 5 .

3.2.20. y (x2 + y2)x2 = y2. pi

(2x+ 2yy)x2 +(x2 + y2

)2x = 2yy

y (2yx2 2y) = 2x3 (x2 + y2) 2xy =

2x3 + 2x (x2 + y2)2y (x2 1) =

x (2x2 + y2)y (x2 1) .

3.2.21. y (3) 3 (x2 + y2)2 = 100xy. pi pi pi

y =25y 3x (x2 + y2)25x+ 3y (x2 + y2) . (3.5)

y (3). . (3.5) x = 3. y (3); 3 (x2 + y2)2 = 100xy x = 3 pi

3 (32 + y2)2 = 100 3 y pi y. y = 1. pi (3, 1)

y (3) =25 1 3 3 (32 + 12)25 3 + 3 1 (32 + 12) =

13

9. (3.6)

3.2.22. y x2 + y2 = 25. pi pi ,

2x+ 2yy = 0 y = xy.

pi pi

y = ddx

(x

y

)= (x)

y xyy2

= y xy

y2=

y x (xy

)y2

= x2 + y2

y3.

.

3. 31

3.3

3.3.1. pi f (x) = c f (x) = 0.

3.3.2. pi (x3) = 3x2.

3.3.3. pi (x2+12x7

). (pi. x27x1

(2x7)2 .)

3.3.4. pi (ex

2 x). (pi. ex2 + 2x2ex2.)

3.3.5. pi (sin (x3 + 7ex)) . (pi. (7ex + 3x2) cos (7ex + x3) .)

3.3.6. pi (cotx). (pi. 1sin2 x

.)

3.3.7. pi

1. (cos(x)) = sinx2. (tan(x)) = tan2 x+ 1

3. (sec(x)) = sinxcos2 x

3.3.8. pi

1. (cosh(x)) = sinh(x),

2. (tanh(x)) = sech2(x),

3. (coth(x)) = csch2(x),4. (sech(x)) = sech(x) tanh(x),5. (csch(x)) = csch(x) coth(x)

3.3.9. pi

1. (arccosx) = 11x2 ,

2. (arctanx) = 11+x2

.

3.3.10. pi

1. (arc cosh(x)) = 1x21 ,

2. (arc tanh(x)) = 11x2 ,

3. (arc coth(x)) = 11x2 ,

4. (arc sech(x)) = 1x

1x2 ,

.

3. 32

5. (arc csch(x)) = 1|x|1+x2 .

3.3.11. pi pi ( ) tan (460) tan (440). (pi. 1. 035 4 0.965 67.)

3.3.12. pi pi ( ) e0.9. (pi. 2. 435 1 .)

3.3.13. y pi pipi

1. x2 + y2 = 16. (pi. x/y.)2. x1/2 + y1/2 = 9. (pi. y/x.)3. x3 xy + y2 = 4. (pi. y3x2

2yx .)

4. x3y3 y = x. (pi. 13x2y33x3y21 .)

5. x3 2x2y + 3xy2 = 38. (pi. 4xy3x23y22x(3yx) .)

3.3.14. y pi pipi

1. x2 + xy = 5 (pi. 10/x3.)

2. x2 y2 = 16 (pi. 16/y3.)3. y2 = x3 (pi. 3x/4y.)

.

4

pi. pi pi .

4.1

4.1.1. f(x) F (x). F (x) f(x)

F (x) = f(x) (4.1)

F (x) =

f(x)dx. (4.2)

4.1.2. pi F (x) f (x) -pi F (x) pi f(x) F (x) pi- f(x).

4.1.3. c f(x)dx = c

f(x)dx

(f(x) + g(x)) dx =

f(x)dx+

g(x)dx

f(u(x))u(x)dx =f(u)du. (4.3)

(4.3) pi pi-:

[f (u (x))] = f (u)u (x) .

33

.

4. 34

4.1.4. pi .1dx = x+ cxmdx = x

m+1

m+1+ c (m 6= 1)

1xdx = ln |x|+ cexdx = ex + csinxdx = cosx+ ccosx)dx = sin x+ ctanx)dx = ln |cosx)|+ csinhx)dx = coshx+ ccosh(xdx = sinh x+ ctanhx)dx = = ln |cosh(x|+ c

4.1.5. c pipi . , F (x) =

f (x) dx, F (x) = f (x). (F (x) + c) = F (x)+(c) =

f (x) + 0, . F (x) + c f (x) ( pi ).

4.1.6. .1

cos(x)dx = ln

1+sinxcosx

+ c1

sin(x)dx = ln

1cosxsinx

+ c1

a2x2dx = arcsin(xa) + c

1a2+x2

dx = 1a

arctan(xa) + c

1x2a2dx =

12a

lnxax+a

+ c1

a2x2dx =12a

lna+xax+ c

1x2a2dx = arccosh(

xa) + c = ln

x+x2 a2+ c1

a2+x2dx = arcsinh(x

a) + c = ln

x+x2 + a2+ c a2 x2dx = 1

2xa2 x2 + a2

2arcsin

(xa

)+ c

a2 + x2dx = 12xa2 + x2 + a

2

2arcsinh

(xa

)+ c

x2 a2dx = 12xx2 a2 a2

2arccosh

(xa

)+ c

a2 + x2dx = 12xa2 + x2 + a

2

2lnx+x2 + a2+ c

x2 a2dx = 12xx2 a2 a2

2lnx+x2 a2+ c

4.2

4.2.1. pi xdx = 1

2x2 + c.

(

12x2 + c

)= 1

22x+ 0 = x.

4.2.2. pi

cosxdx = sinx+ c. (sinx+ c) = cosx+ 0 = cosx.

4.2.3. pi c f(x)dx = c f(x)dx.

, F (x) =f(x)dx, . F (x) = f (x). (c F (x)) =

c F (x) = c f (x), . c F (x) c f (x).

.

4. 35

4.2.4. pi x4dx.

x4dx = 1

5x5 + c.

4.2.5. pi

(2 5 sinx+ ex) dx

(2 5 sinx+ ex) dx = 2dx 5 sinxdx+ exdx = 2x+ 5 cosx+ ex + c.

4.2.6. pi

2x+x+1x

dx

2x+x+ 1x

dx =

2x1/2dx+

dx+

x1/2dx

= 2x3/2

3/2+ x+

x1/2

1/2=

4

3x3/2 + x+ 2x1/2 + c.

4.2.7. pi

xxxdx.

x

xxdx =

xx x1/2dx =

xx3/2dx

=

x x3/4dx =

x7/8dx =

8

15x15/8 + c.

4.2.8. pi 2x+ 3

5

x

33x2

dx. 2x+ 3

5

x

33x2

dx =2

3

x

x2/3dx+

1

5

x1/2

x2/3dx =

2

3 34x4/3 +

1

5 65x5/6 =

1

2x4/3 +

6

25x5/6 + c.

4.2.9. pi

(x 1)2 3xdx.

(x 1)2 3xdx = (x 2x1/2 + 1) x1/3dx

=

(x4/3 2x2/3 + x1/3) dx = 3

7x7/3 6

5x5/3 +

3

4x4/3 + c.

4.2.10. pi

cot2 xdx.cot2 xdx =

cos2 x

sin2 xdx =

1 sin2 x

sin2 xdx =

1

sin2 xdx

dx = cotx x+ c.

4.2.11. pi

3

1 2xdx. u = 1 2x, du = 2dx,

3

1 2xdx = 12

u1/3du = 3

8u4/3 = 3

8(1 2x)4/3 + c.

.

4. 36

4.2.12. pi

x(x2+1)

x2+1

dx. u = x2 + 1, du = 2xdx,

x

(x2 + 1)x2 + 1

dx =1

2

du

u3/2=

1

2

u3/2du =

1

2(2

1u1/2

)= 1

x2 + 1+ c.

4.2.13. pi e3x+2dx

u = 3x+ 2, du = 3dx, e3x+2dx =

1

3

eudu =

1

3eu =

1

3e3x+2 + c.

4.2.14. pi

x(2x+5)2

dx.

u = 2x+ 5, du = 2dx, x

(2x+ 5)2dx =

1

2

(u 5) /2

u2du =

1

4

u 5u2

du =1

4

du

u 5

4

du

u2

=1

4ln |u| 5

4 u1

1 =1

4ln |2x+ 5|+ 5

4 1

2x+ 5+ c.

4.2.15. pi xx 3dx.

u = x 3, du = dx, xx 3dx =

(u+ 3)u1/2du =

(u3/2 + 3u1/2

)du

=2

5u5/2 + 3 2

3 u3/2 = 2

5(x 3)5/2 + 2 (x 3)3/2 + c.

4.2.16. pi

cos5 x sinxdx. u = cosx, du = sinxdx, pi

cos5 x sinxdx = u5du = 1

6u6 = 1

6cos6 x.

4.2.17. pi

lnxxdx.

u = lnx, du = dx/x, pilnx

xdx =

udu =

1

2u2 =

1

2(lnx)2 + c.

4.2.18. pi x+1/2

x2+x+3dx.

u = x2 + x+ 3, pi du = (2x+ 1) dx. x+ 1/2

x2 + x+ 3dx =

1

2

du

u=

1

2ln |u| = 1

2lnx2 + x+ 3+ c.

.

4. 37

4.2.19. pi

dxx2+4x+5

. , pi ,

x2 + 4x+ 5 = x2 + 2 2 x+ 22 + 1 = (x+ 2)2 + 1.pi

dx

x2 + 4x+ 5=

d (x+ 2)

(x+ 2)2 + 1= arctan (x+ 2) + c.

4.2.20. pi

2x35x212x+52x+3

dx. pi pi 2x3 5x2 12x + 5

2x+ 3. 2x3 5x2 12x +5 2x+ 32x3 +3x2 x2 4x

8x2 12x +58x2 12x

+5

pi pi x2 4x pipi 5, .2x3 5x2 12x+ 5

2x+ 3= x2 4x+ 5

2x+ 3.

pi 2x3 5x2 12x+ 5

2x+ 3dx =

(x2 4x+ 5

2x+ 3

)dx =

x3

3 2x2 + 5

2ln |2x+ 3|+ c.

4.3

4.3.1. pi x2dx = 1

3x3 + c.

4.3.2. pi

1dx = x+ c.

4.3.3. pi

xdx = 32x3/2 + c.

4.3.4. pi

sin dx = cosx+ c.

4.3.5. pi

1xdx = lnx+ c.

4.3.6. pi

(f(x) + g (x)) dx =f(x)dx+

g(x)dx.

4.3.7. pi x3dx. (pi. 1

4x4 + c.)

4.3.8. pi

(2x+ 4)dx. (pi. x2 + 4x, u = 2x+ 4, du = 2dx).

4.3.9. pi

x3x4+5

dx. (pi. 12

(x4 + 5), u = x4 du = 4x3dx).

4.3.10. pi

x2+2(x3+6x+1)

dx. (pi. 13

ln (x3 + 6x+ 1), u = x3 + 6x + 1, du =(3x2 + 6)dx).

.

4. 38

4.3.11. pi

dx4x1 . (pi.

14

ln (4x 1), u = 4x 1, du = 4dx).

4.3.12. pi

(ex + 2)2exdx. (pi. 13

(ex + 2)3, u = ex, du = exdx).

4.3.13. pi sin(2x)+cos(2x)

cos(2x)dx.. (pi.

sin(2x)cos(2x)

dx+ cos(2x)

cos(2x)dx = 1

2ln (cos 2x) +

x, u = cos(2x), du = 2 sin(2x)dx 2 ).4.3.14. pi

1

9x2dx. (pi. arcsin13x, u = x

3, du = 1

3dx).

4.3.15. pi

194x2dx. (pi.

12

arcsin 23x, u = 2

3x, du = 2

3dx).

4.3.16. pi

14x2+25

dx. (pi. 110

arctan 25x, u = 2

5x, du = 2

5dx).

4.3.17. pi

110+4xx2dx. (pi. arcsin

114

14 (x 2), pi

).

4.3.18. pi

x+454xx2dx. (pi.

(5 4x x2) + 2 arcsin (1

3x+ 2

3

), u =

x+ 3, pi pi ).

.

5

pi pi pi .

5.1

5.1.1. pi pi f(u(x))u(x)dx =

f(u)du.

.

5.1.2. .

1. a2 + b2x2 pi x = a

btan(u) pi a

1 + tan2(u) =

acos(u)

.

2. a2 b2x2 pi x = a

bsin(u) pi a

1 sin2(z) =

a cos(u).

3. b2x2 a2 pi x = a

b1

cos(u) pi a

1

sin2(u) 1 =

a tan(u).

4. b2x2 a2 pi x = a

bcosh(u) pi a

cosh2(u) 1 =

a sinh(u).

5.1.3. pi pi: f(x) g(x) ,

(f(x)g(x)) = f (x)g(x) + f(x)g(x)(f(x)g(x)) dx =

f (x)g(x)dx+

f(x)g(x)dx

f (x)g(x)dx = f(x)g(x)f(x)g(x)dx.

39

.

5. 40

5.1.4. pi pipi :(fdg + gdf) =

d (fg) = fg.

5.1.5. pi pi pi .

1.x cosxdx = cosx+ x sinx+ c.

2.x sinxdx = sinx x cosx+ c.

3.xexdx = xex ex + c.

5.1.6. pipi pi pi

A

x x0 ,A

(x x0)2, ... (5.1)

A

ax2 + bx+ c,

A

(ax2 + bx+ c)2, ... (5.2)

Ax+ b

ax2 + bx+ c,

Ax+ b

(ax2 + bx+ c)2, .... (5.3)

5.1.7. : (5.2) (5.3) b2 4ac 0 (5.1). pipi b2 4ac < 0 .5.1.8. pi pi . - pi (pi E =

4ac b2):

A

x x0dx = A ln |x x1|A

(x x0)2dx = A

x x0A

ax2 + bx+ cdx =

2A

Earctan

2ax+ b

EA

(ax2 + bx+ c)2dx =

A (2ax+ b)

E2 (ax2 + bx+ c)+

4Aa

E3arctan

2ax+ b

EAx+B

ax2 + bx+ cdx =

A

2aln(ax2 + bx+ c

)+

2B

E

(arctan

2ax+ b

E

) AE ba

(arctan

2ax+ b

E

)5.1.9. P (x) Q(x) pi, f(x) = P (x)

Q(x) .

5.1.10. pi pi , pi pi .

5.1.11. pi P (x)/Q(x) P (x) pi Q(x). x0 Q(x). pipi.

.

5. 41

1. pi pi, pi P (x)/Q(x) -

A

x x0 .

2. pi pipi n, pi P (x)/Q(x) n

A1x x0 ,

A2

(x x0)2, ...,

An(x x0)n .

3. x0 pi, x0 pi Q(x) (x x0)(x x0) ax2 + bx+ c pi a, b, c pi . pi P (x)/Q(x)

Ax+B

ax2 + bx+ c.

4. , x0 pipi n, pi P (x)/Q(x) n

Ax+B

ax2 + bx+ c,

Ax+B

(ax2 + bx+ c)2, ...,

Ax+B

(ax2 + bx+ c)n

5.1.12. pi pipi - . , pipi f (x) P (x) pi Q (x) pi pi .

5.1.13. P (x) Q(x), pi pi

f(x) = P1(x) +P2(x)

Q(x)

pi P1 (x), P2 (x) pi P2 (x) pi Q (x). pi pi f (x).

5.2

5.2.1. pi

(1 2x)100 dx. u = 1 2x, du = 2dx

(1 2x)100 dx = 12

(1 2x)100 d (1 2x) = 1

202u101 + c = 1

202(1 2x)101 + c.

.

5. 42

5.2.2. pi

16 x2dx. sinu = x

4, cosudu = dx

4.

16 x2dx = 4

1 sin2 x4 cosudu = 16

cos2 udu = 16

1 + cos 2u

2du

= 8

du+ 8

cos 2udu = 8u+ 4 sin (2u) .

, u = arcsin x4. pi, sin (2u) = 2 sinu cosu = 2x

4

1 x2

16. pi

16 x2dx = 8 arcsin x4

+x

2

16 x2 + c.

5.2.3. pi

dx3x28x+5 .

pi

3x2 8x+ 5 = 3(x 4

3

)2 1

3.

pi ( u = x 43)

dx

3x2 8x+ 5 =

du

3u2 13

=

13

u2 (13

)2du = 32 (

1

3u 1 1

3u+ 1

)dt

=1

2ln

u 13 12 ln

u+ 13 = 12 ln

3x 53x 3+ c

5.2.4. pi

1+ex

1exdx u = ex, du = exdx

1 + ex

1 exdx =

1 + u

1 u 1

udu =

(2

1 u 1

u

)du = 2 ln |1 u|+ ln |u|

= 2 ln |1 ex|+ ln ex = 2 ln |1 ex|+ x+ c.

5.2.5. pi

2x7x27x+1dx.

2x 7

x2 7x+ 1dx =

(x2 7x+ 1)x2 7x+ 1 dx ==

d (x2 7x+ 1)x2 7x+ 1 = ln

x2 7x+ 1+ c.5.2.6. pi

2x sinxx2 cosx

sin2 xdx.

2x sinx x2 cosx

sin2 xdx =

(x2)

sinx x2 (sinx)

sin2 xdx =

(x2

sinx

)dx =

x2

sinx+ c

.

5. 43

5.2.7. pi

dx1x2(1+x) .

cosu = x, sinudu = dx. dx

1 x2 (1 + x) = sinudu

sinu (1 + cosu)=

du

1 + cosu=

du

2 cos2 u2

= 12

tanu

2

=

1 cosu1 + cosu

=

1 x1 + x

+ c.

5.2.8. pi

sin3 x cos2 xdx. u = cosx, du = sinxdx.

sin3 x cos2 xdx =

sin2 x cos2 x sinxdx =

(1 u2)u2du

= (

u2 u4) du = u33

+u5

5

= cos3 x

3+

cos5 x

5+ c.

5.2.9. pi

sin2 x cos2 xdx.

sin2 x cos2 xdx =1

4

(sin2 (2x)

)dx =

1

4

1 cos (4x)

2dx

=1

8x 1

2

cos (4x) dx =

x

8 1

32sin (4x) + c.

5.2.10. pi

sinx cos3 xdx. u = cosx, du = sinxdx.

sinx cos3 xdx =

cos3 xd (cosx) = u3du = 1

4u4 = 1

4cos4 x+ c.

5.2.11. pi

sin2 x cos4 xdx.

sin2 cos4 xdx =

(sinx cosx)2 cos2 xdx =

1

4(sin 2x)2

(1 + cos 2x

2

)dx

=1

8

sin2 (2x) dx+

1

8

sin2 (2x) cos (2x) dx

=1

8

1 cos (4x)

2dx+

1

16

sin2 (2x) d (sin (2x))

=1

16x 1

64sin (4x) +

1

48sin3 (2x) + c.

5.2.12. pi

x cosxx sinx+cosx

dx. u = x sinx+ cosx, du = (sinx+ x cosx sinx) dx.

x cosx

x sinx+ cosxdx =

du

u= lnu = ln (x sinx+ cosx) + c.

.

5. 44

5.2.13. pi

sin7 xdx. u = cosx, du = sinxdx

sin7 xdx =

sin6 x sinxdx =

(1 cos2 x)3 d (cosx) = (1 u2)3 du

= (

1 3u2 + 3u4 u6) du = u+ u3 35u5 +

u7

7

= cosx+ cos3 x 35

cos5 x+ cos7 x+ c.

5.2.14. pi

sin4 xdx.

sin4 xdx =

(sin2 x

)2dx =

(1 cos (2x)

2

)2dx =

(1

4 cos (2x)

2+

cos2 (2x)

4

)dx

=x

4 1

4sin (2x) +

1

4

1 + cos (4x)

2dx =

x

4 1

4sin (2x) +

x

8+

1

32sin (4x) + c.

5.2.15. pi

xdxx+1 3x+1 .

u = (x+ 1)1/6 u6 = x+ 1, 6u5du = dx. xdx

x+ 1 3x+ 1 = 6

(u6 1)u5u3 u2 du = 6

(u6 1)u5u2 (u 1) du

= 6

u3(u5 + u4 + u3 + u2 + u+ 1

)du

= 6

(u9

9+u8

8+u7

7+u6

6+u5

5+u4

4

)= 6

((x+ 1)9/6

9+

(x+ 1)8/6

8+

(x+ 1)7/6

7+

(x+ 1)

6+

(x+ 1)5/6

5+

(x+ 1)4/6

4

).

5.2.16. pi x cosxdx pi.

f = sinx, g = x, x cosxdx = x sinx

sinxdx = x sinx+ cosx+ c.

5.2.17. pi xexdx

f = ex, g = x, xexdx = xex

exdx = xex ex + c.

5.2.18. pi x2exdx.

f = x3, g = ex, x2exdx = x3ex 3

x2exdx = x3ex 3

(x2ex

exdx2

)= x3ex 3x2ex + 3

2xexdx = x3ex 3x2ex + 6

(xex

exdx

)= x3ex 3x2ex + 6xex 6ex + c.

.

5. 45

5.2.19. pi x ln (1 + x2) .

f = x2/2, g = ln (1 + x2), x ln

(1 + x2

)dx =

x2

2ln(1 + x2

) x22

2x

1 + x2dx =

x2

2ln(1 + x2

) x2/21 + x2

d(x2)

=x2

2ln(1 + x2

) x22

+1

2ln(1 + x2

)+ c.

5.2.20. pi

sin2 xdx. sin2 xdx =

sin(x)d(cos(x))

= sin(x) cos(x) +

cos(x)d(sin(x)) = sin(x) cos(x) +

cos2(x)dx

= sin(x) cos(x) +

(1 sin2(x))dx = sin(x) cos(x) + x

sin2(x)dx.

sin2(x) = sin(x) cos(x) + x

sin2(x)dx 2

sin2(x) = sin(x) cos(x) + x

sin2(x) = sin(x) cos(x) + x

2.

5.2.21. pi x arctanxdx.

f = x22, g = arctanx,

x arctanxdx =x2

2arctanx

x2

2d (arctanx) =

x2

2arctanx

x2

2

1

1 + x2dx

=x2

2arctanx 1

2

x2 + 1 1x2 + 1

dx =x2

2arctanx 1

2

dx+

1

2

dx

x2 + 1

=x2

2arctanx 1

2x+

1

2arctanx+ c.

5.2.22. pi eax sin (bx) dx.

pi J1 =eax sin (bx) dx J2 =

eax cos (bx) dx.

J1 =

eax sin (bx) dx =

1

2eax sin (bx)x 1

a

eax (sin (bx)) dx

=1

aeax sin (bx) b

a

eax cos (bx) dx =

1

aeax sin (bx) b

aJ2.

J2 =

eax cos (bx) dx =

1

2eax cos (bx)x+

1

a

eax (cos (bx)) dx

=1

aeax cos (bx) +

b

a

eax sin (bx) dx =

1

aeax cos (bx) +

b

aJ1.

.

5. 46

.

J1 +b

aJ2 =

eax

asin (bx)

baJ1 + J2 =

eax

acos (bx) .

pi J1, J2 pi

J1 = beax cos bx aeax sin bx

a2 + b2, J2 =

aeax cos bx+ beax sin bx

a2 + b2

5.2.23. pi

(lnx)2 dx.

f = x, g = (lnx)2, (lnx)2 dx = x (lnx)2

xd((lnx)2

)= x (lnx)2

2x (lnx) 1

xdx

= x (lnx)2 2

lnxdx = x (lnx)2 2 (x lnx

xd (lnx)

)= x (lnx)2 2

(x lnx

x

1

xdx

)= x (lnx)2 2x lnx+ 2x+ c.

5.2.24. pi

1x21dx.

1

x2 1 =A

x 1 +B

x+ 1=

(A+B) x+ (AB)x2 1 .

A+B = 0AB = 1

}{A =

1

2, B = 1

2

}

1

x2 1dx =1

2

1

x 1 1

2

1

x+ 1=

1

2(ln |x 1| ln |x+ 1|) = 1

2ln

x 1x+ 1+ c.

5.2.25. pi

x210x+13(x1)(x2)(x3)dx.

x2 10x+ 13(x 1) (x 2) (x 3) =

A

x 1 +B

x 2 +C

x 3=A (x 2) (x 3) +B (x 1) (x 3) + C (x 1) (x 2)

(x 1) (x 2) (x 3)=

(A+B + C)x2 (5A+ 4B + 3C)x+ (6A+ 3B + 2c)(x 1) (x 2) (x 3) .

.

5. 47

A+B + C = 1

5A+ 4B + 3C = 10

6A+ 3B + 2C = 13

A = 2, B = 3, C = 4 x2 10x+ 13

(x 1) (x 2) (x 3)dx =

2dx

x 1 +

3dx

x 2

4dx

x 3= 2 ln |x 1|+ 3 ln |x 2| 4 ln |x 3|+ c.

5.2.26. pi

x2x2x3+x26xdx.

x2x2 = (x+ 1) (x 2) x3 +x26x = x (x+ 3) (x 2). pix2 x 2x3 + x2 6xdx =

x+ 1

x (x+ 3)dx.

pix+ 1

x (x+ 3)=A

x+

B

x+ 3=

(A+B)x+ 3A

x (x+ 3)

pi A+B = 1, 3A = 1 A = 1/3, B = 2/3 x2 x 2x3 + x2 6xdx =

1

3

dx

x+

2

3

dx

x+ 3=

1

3ln |x|+ 2

3ln |x+ 3|+ c.

5.2.27. pi

dxx4x3+x2 .

x4 x3 + x2 = x2 (x2 x+ 1)

x2 x+ 1 pi . pi 1

x4 x3 + x2 =A

x+B

x2+

Cx+D

x2 x+ 1 =(A+ C)x3 + (A+B +D)x2 + (AB)x+B

x2 (x2 x+ 1) .

pi

A+ C = 0

A+B +D = 0AB = 0

B = 1

pi A = 1, B = 1, C = 1, D = 0. dx

x4 x3 + x2 =dx

x+

dx

x2

xdx

x2 x+ 1= lnx 1

x 1

2ln(x2 x+ 1) 3

3arctan

3

(2

3x 1

3

)+ c.

.

5. 48

5.2.28. pi

x2

(x2+1)(x2+x+1)dx.

x2

(x2 + 1) (x2 + x+ 1)=Ax+B

x2 + 1+

Cx+D

x2 + x+ 1

pi pi pipi

A+ C = 1

A+B +D = 1

A+B + C = 0

B +D = 0

, A = 1, B = 0, C = 1, D = 0, .x2

(x2 + 1) (x2 + x+ 1)dx =

x

x2 + 1dx

x

x2 + x+ 1dx

=1

2ln(x2 + 1

) 12

ln(x2 + x+ 1

)+

3

3arctan

3

(2

3x+

1

3

)+ c.

5.2.29. pi

x43x33x2+10(x+1)2(x3) dx.

(x+ 1)2 (x 3) = x3 x2 5x 3, pi, pipi pi .

x4 3x3 3x2 +0x +10 x3 x2 5x 3x4 x3 5x2 3x x 22x3 +2x2 +3x +102x3 +2x2 +10x +6

7x +4pi pi x 2 pipi 7x+ 4, .

x4 3x3 3x2 + 10(x+ 1)2 (x 3) dx =

(x 2 + 7x+ 4

(x+ 1)2 (x 3)

)dx

=x2

2 2x

7x 4

(x+ 1)2 (x 3)dx.

7x 4

(x+ 1)2 (x 3) =A

x+ 1+

B

(x+ 1)2+

C

(x 3)

=(A+ C)x2 + (2A+B + 2C)x+ (3A 3B + C)

(x 3) (x+ 1)2pi

A+ C = 0

2A+B + 2C = 73A 3B + C = 4

.

5. 49

A = 1716, B = 11

4, C = 17

16

7x 4(x+ 1)2 (x 3)dx =

(17/16x+ 1

+11/4

(x+ 1)2+

17/16

x 3)dx

= 114 (x+ 1)

1716

ln (x+ 1) +17

16ln (x 3) .

x2

2 2x+ 11

4 (x+ 1)+

17

16ln (x+ 1) 17

16ln (x 3)

5.2.30. pi

dxx+

2x+3.

u =

2x+ 3, udu = 2dx. dx

x+

2x+ 3=

1

2

udu

u232

+ u=

u

u2 + 2u 3du

=

(1

2 (u 1) +3

2 (u+ 3)

)du =

1

2ln |u 1|+ 3

2ln |u+ 3|

=1

2ln(

2x+ 3 1)

+3

2ln(

2x+ 3 + 3)

+ c.

5.3

5.3.1. pi

e1/x

x2dx (pi. e 1x , u = 1

x).

5.3.2. pi

1e2x+1

dx (pi. 12

ln (e2x + 1) + 12

ln (e2x), u = e2x + 1, du =2exdx).

5.3.3. pi

cos2(x) sin(x)dx (pi. 13

cos3 x, u = cosx, du = sinxdx).5.3.4. pi

e2x cos(e2x)dx (pi. 1

2sin (e2x), u = e2x, du = 2e2xdx).

5.3.5. pi

11+cos(x)

dx (pi. tan 12x, cosx = 1tan

2 x2

1+tan2 x2).

5.3.6. pi

14(x+2)2dx (pi. arcsin

(12x+ 1

), u = 1

2x, du = 1

2dx).

5.3.7. pi

14xx2dx (pi. arcsin(

12x+ 1

), pipi).

5.3.8. pi

dxe2x+e2x (pi.

12

arctan (e2x), u = e2x, du = 2e2xdx).

5.3.9. pi

19x24dx (pi.

112

ln (3x 2) 112

ln (3x+ 2)).

5.3.10. pi

x+3x2+4

dx (pi.x2 + 4 + 3 arcsinh 1

2x, u = x2 + 4,

pi ).

5.3.11. pi

cos3 xdx (pi. 13

cos2 x sinx+ 23

sinx, u = sinx).

.

5. 50

5.3.12. pi

sin2 x cos2 xdx(pi. 1

4sinx cos3 x+ 1

8cosx sinx+ 1

8x, sin2 x cos2 x = sin2 2x).

5.3.13. pi

sin 2x cos 4xdx (pi. 112

cos 6x + 14

cos 2x, sin 2x cos 4x =12

sin 6x 12

sin 2x).

5.3.14. pi

sinh x2dx (pi. 2 cosh 1

2x.)

5.3.15. pi ex sinhxdx (pi.1

2coshx sinhx 1

2x + 1

2cosh2 x, sinhx =

exex2

).

5.3.16. pi

19x24dx (pi.

13

ln(

3x+

(9x2 4)), u = 3

2x).

5.3.17. pi

4x2 + 9dx (pi. 12x

(4x2 + 9)+94

arcsinh 23x, u = arcsinh 2

3x).

5.3.18. pi

19x24dx (pi.

112

ln (3x 2) 112

ln (3x+ 2), u = 32x).

5.3.19. pi

1x2

9+x2dx (pi. 1

9x

(9 + x2), x = 3 tanu, dx = (1+ x2

9)du).[

d(ab

tan(u))

du= a

b(1 + tan2 u)

]5.3.20. pi

x2x29dx (pi.

12x

(x2 9) + 92

ln(x+

(x2 9)

), x =

2cos(z)

, dx = 13

tan(z)cos(z)

dz).

5.3.21. pi

49x2x

dx (pi.

(4 9x2) 2 arctanh 12

(4 9x2), x =

23

sin(z), dx = 23

cos zdz).

5.3.22. pi

1x

9+4x2dx (pi. 1

3arctanh 1

3

(9 + 4x2), x = 3

2tan(u), dx =

32 cos2(z)

dz).

5.3.23. pi

x22xx2dx (pi. 12x

(2x x2) 3

2

(2x x2)+ 3

2arcsin (x 1),

x 1 = sin(z), dx = cos(z)dz).5.3.24. pi

1

x

42xdx (pi. arctanh 12

(4 2x), 4 2x = z2, dx =1

2zdz).

5.3.25. pi

1(x1)x+3dx (pi. arctanh 12

(x+ 3), x + 3 = u2, dx =

2udu).

5.3.26. pi

1x1/3x2/3dx (pi. 3 3

x2 ln ( 3x 1)+ln

(( 3x)

2+ 3x+ 1

)

ln (x 1), x = u3, dx = 3u2du).5.3.27. pi

1

1+sin(x)cos(x)dx (pi. ln(tan 1

2x) ln (tan 1

2x+ 1

), sin(x) =

2u1+u2

, dx = 21+u2

dz).

5.3.28. pi

13+sin(x)

dx (pi. 12

2 arctan 1

8

(6 tan 1

2x+ 2

)2).

.

5. 51

5.3.29. pi x sinxdx (pi. sinx x cosx).

5.3.30. pi x3 lnxdx (pi.

ln(x)d(1

4x4) = 1

4x4 ln(x) 1

4

x4d(ln(x)) =

14x4 ln(x) 1

4

x4 1

xdx = 1

4x4 lnx 1

16x4).

5.3.31. pi

arcsinxdx (pi. x arcsinx+

1 x2).5.3.32. pi

x2 sinxdx (pi. x2 cosx+ 2 cosx+ 2x sinx).

5.3.33. pi

tan1 xdx (pi. x arctanx 12

ln (1 + x2)).

5.3.34. pi x2e2xdx (pi. 1

2x2e2x 1

2xe2x 1

4e2x).

5.3.35. pi

cos(ln(x))dx (pi. 12x (sin (lnx) + cos (lnx))).

5.3.36. pi

1(9+x2)3/2

dx (pi. 19

x(9+x2)

).

5.3.37. pi

x21+x

dx (pi. 25

(x+ 1

)5 43

(x+ 1

)3+ 2x+ 1).

5.3.38. pi

1x29dx (pi.

(16

ln (x 3) 16

ln (x+ 3))).

5.3.39. pi

x+1x2+5x+6

dx (pi. (

2x+3 1

x+2

)dx = 2 ln (x+ 3) ln (x+ 2)).

5.3.40. pi

x+2x3x2x+1dx (pi.

(3

2(x1)2 14(x1) + 14(x+1))dx = 1

4ln (x+ 1)

32(x1) 14 ln (x 1)).

5.3.41. pi

x+1(x+2)3

dx (pi. ( 1

(x+2)3+ 1

(x+2)2

)dx = 1

2(x+2)2 1

x+2).

5.3.42. pi

x2+x+3x2+x32x2dx (pi.

(9

5(x2) 15 3+4x1+x2)dx= 9

5ln (x 2)2

5ln (1 + x2)

35

arctanx).

5.3.43. pi

1x2+7x+6

dx (pi. 15

ln (x+ 6) + 15

ln (x+ 1)).

5.3.44. pi

x2+3x4x22x8dx (pi. x+ ln (x+ 2) + 4 ln (x 4)).

5.3.45. pi

1x3+x

dx (pi. lnx 12

ln (1 + x2)).

5.3.46. pi

x3+x1(x2+1)2

dx (pi. 12

xx2+1 1

2arctanx+ 1

2ln (x2 + 1)).

5.3.47. pi

1e2x4exdx (pi.

14ex 1

16ln (ex) + 1

16ln (ex 4)).

.

6

f (x) pi a b pipi ( pi f (x), a, b). pi pi.

6.1

6.1.1. f(x) 0, pi X R.pi a, b ( a b) pi pi f (x), x x = a, x = b ( 6.1). F (b) F (a), pi F (x) pi pi F (x) = f (x)( F (x) =

f (x) dx).

6.1.2. f(x) X R ( pi pi pi pi ). a, b X, f(x)

baf(x) : b

a

f(x) = F (b) F (a)

pi F (x) pipi pi F (x) = f(x) ( F (x) =f (x) dx).

6.1.3. 6.1.2, baf(x) = F (b)F (a) pi

pi pi f (x), x x = a, x = b, pi pi pi x ( pi, pi).

6.1.4. ( f (x), g (x) - ):

1. bac f(x)dx = c b

af(x)dx

52

.

6. 53

2. ba

(f(x) + g(x)) dx = baf(x)dx+

bag(x)dx

3. baf(x) +

cbf(x) =

caf(x)dx

4. (x [a, b] : 0 f(x)) 0 baf(x)dx

5. (x [a, b] : g(x) f(x)) bag(x)d b

af(x)dx

6. (x [a, b] : A f(x) B) A (b a) baf(x)dx B (b a)

7. x0 [a, b] : f(x0) = 1ba baf(x)dx

6.1.5. 1 pi pi pi.

1. a = b

.

6. 54

6.2

6.2.1. 6.1.1. 6.1. x0, x1, ...., xN pi a = x0 < x1 1, = n 1).6.3.22.

0exdx (pi. 1).

6.3.23.

0xex

2dx (pi. 1/2).

6.3.24.

11

1+x2dx (pi. pi/4).

6.3.25.

11

x2x21dx (pi. 1).

6.3.26.

11

x2+xdx (pi. ln(2)).

6.3.27.

0x2ex/2dx (pi. 16).

6.3.28. 6

2

[1

(4x)2]1/3

dx (pi. 621/3).

6.3.29. 2

01

(x1)2dx (pi. ).

6.3.30. 2

01

(x1)3dx (pi. 0).

6.3.31. 3

21

(x1)3dx (pi. 3/8).

.

7

pi pi pi / / .

7.1

7.1.1. f (x) pi pi [a, b]. (. pi) f (x) pi a b

s =

ba

1 +

(df

dx

)2dx. (7.1)

7.1.2. pi pi pi pi pi f1(x), f2 (x) x = a x = b. pi pi pi pi A R2.7.1.3. pipi pi pi pi pi . (x, y) ( ) pi pi pi :

x =

bax (f2(x) f1 (x)) dx ba

(f2(x) f1 (x)) dx, y =

12

ba

[(f2(x))

2 (f1 (x))2]dx b

a(f2(x) f1 (x)) dx

. (7.2)

7.1.4. pipi pi pi pi f (x), x x = a x = b (x, y) pi pi :

x =

baxf(x)dx baf(x)dx

, y =12

ba

(f(x))2 dx baf(x)dx

(7.3)

(. (7.2) f1 (x) = 0, f2 (x) = f (x)).

67

.

7. 68

7.2

7.2.1. (7.1). 7.1. [a, b] -

7.1: .

[x0, x1], [x1, x2], ..., [xN1, xN ], pi a = x0 < x1 < ... < xN = b

x1 x0 = x2 x1 = ... = xN xN1 = x = b aN

.

pi pi pi pi A0A1...AN pi x, . N . n- A0A1...AN , . An1An.

sn =

x2 + f 2n = x

1 +

(fnx

)2.

N

sN =Nn=1

sn =Nn=1

1 +

(fnx

)2x

pi pi

s = limN

sN = limN

Nn=1

1 +

(fnx

)2x =

ba

1 +

(df

dx

)2dx

pi pi pi , fn

x pi df

dx.

.

7. 69

pipi , pi -pi pi (pi pi pi).7.2.2. pi pi f (x) = 1

2x2 + 3, a = 0, b = 1.

pi (7.1),

s =

x2x1

1 +

(df

dx

)2dx =

10

1 + x2dx =

[x

2+

1

2log(x+

1 + x2)]x=1

x=0

=1

2

2 +

1

2log(

1 +

2).

7.2.3. pi pi f (x) = x48

+ 14x2

, x1 = 1, x2 = 4.

df

dx=x3

2 1

2x3pi

s =

41

1 +

(x3

2 1

2x3

)2dx =

41

1 +

x6

4+

1

4x6 1

2dx =

41

(x3

2+

1

2x3

)2dx

=

41

(x3

2+

1

2x3

)dx =

2055

64.

7.2.4. pi pi f (y) = y42

+ 116y2

, a = 1, b = 2. pi y. pi

df

dy= 2y3 1

8y3

s =

ba

1 +

(df

dy

)2=

21

1 +

(2y3 1

8y3

)2dy =

21

1 +

(2y3 +

1

8y3

)2dy =

483

64

7.2.5. pi pi pi pipi : x2/3+y2/3 = a2/3.

7.2. x

2

3x1/3 +

2

3y1/3

dy

dx= 0 dy

dx= x

1/3

y1/3.

pi x pi a a. pi

s =

aa

1 +

(x1/3

y1/3

)2dx =

aa

x2/3 + y2/3

x2/3dy =

aa

a2/3

x2/3dy

= a1/3 aax1/3dy =

3

2a1/3

[x2/3

]x=ax=a = 3a.

pi. pi, . 6a.

.

7. 70

7.2: pi 7.2.5.

7.2.6. pi (7.3) pipi pi

pipi pi pi pi, pi pi x f (x). pi, x. , (x, y) , (x, y), pi pi y - pi ( pi y) pi pipi. n- pi ( pi , pi ) mn = f (xn) x pi pi y xn mn = xn f (xn) x. pi pi

Nn=1 xn f (xn) x

pipi pi (x, y), pi x Nn=1 f (xn) x. pipi

x Nn=1

f (xn) x =Nn=1

xn f (xn) x

pi N

x ba

f (x) dx =

ba

xf (x) dx x = baxf (x) dx baf (x) dx

.

pi pi pi pi (7.3). pi, pi pi x

y Nn=1

f (xn) x =Nn=1

1

2f (xn) f (xn) x

.

7. 71

pi

y ba

f (x) dx =1

2

ba

[f (x)]2 dx y = ba

[f (x)]2 dx baf (x) dx

.

7.2.7. pi pi pi pi f (x) = 4 x2 x = 0 y = 0.

f (x) x x = 2. pi

x =

20xf (x) dx 2

0f (x) dx

=

20x (4 x2) dx 20

(4 x2) dx

y =12

20

[f (x)]2 dx 20f (x) dx

=

20

(4 x2)2 dx 20

(4 x2) dx pi 2

0

(4 x2) dx = [4x x3

3

]x=2x=0

=16

3, 2

0

x (4 x2) dx = [2x2 x44

]x=2x=0

= 4, 20

(4 x2)2 dx = 2

0

(16 + x4 8x2) dx = [16x+ x5

5 8x

3

3

]x=2x=0

=256

15.

pix =

4

16/3=

3

4, y =

12256/15

16/3=

8

5.

7.2.8. pi pi pi pi f1 (x) = 2 sin 3x y = 0.

x =

pi/30

xf1 (x) dx pi/30

f1 (x) dx=

pi/30

x 2 sin 3xdx pi/30

2 sin 3xdx

y =12

pi/30

[f1 (x)]2 dx pi/3

0f1 (x) dx

=12

pi/30

(2 sin 3x)2 dx pi/30

2 sin 3xdx

pi/30

2 sin 3xdx =4

3, pi/3

0

x 2 sin 3xdx = 29pi, pi/3

0

(2 sin 3x)2 dx =2

3pi.

.

7. 72

pix =

2pi/9

4/3=pi

6, y =

pi/3

4/3=pi

4.

7.2.9. pi pi pi pi f1 (x) = x1/2, f2 (x) = x2/8.

f1 (x) , f2 (x) (0, 0) (4,2). pi

x =

20x (f2 (x) f1 (x)) dx 4

0(f2 (x) f1 (x)) dx

=

40x (x2

8+ x1/2

)dx 4

0

(x28

+ x1/2)dx

y =12

20

([f2 (x)]

2 [f1 (x)]2)dx 2

0(f2 (x) f1 (x)) dx

=

12

40

(x4

64 x)dx 4

0

(x28

+ x1/2)dx

40

(x

2

8+ x1/2

)dx =

8

3, 4

0

x (x

2

8+ x1/2

)dx =

24

5, 4

0

(x4

64 x)dx = 24

5.

pix =

24/5

8/3=

9

5, y =

12/58/3

= 910.

7.2.10. pi - pi pi pi f1 (x) =

4 x2, f2 (x) =

4 x2

x = 0 x =

2. 7.3. , pi y = 0. pi

x =

20

x (f2 (x) f1 (x)) dx 20

(f2 (x) f1 (x)) dx=

20

2x

4 x2dx 20

2

4 x2dx.

pi 20

2

4 x2dx = 4 2

0

1 x

2

4dx = pi + 2

20

2x

4 x2dx = 4 2

0

x

1 x

2

4dx =

16

3 4

3

2

pi

x =163 4

3

2

pi + 2

.

7. 73

7.3: pi 7.2.10.

.

7. 74

7.3

pi pi pipi pi .

7.3.1. y2 = x3 pi pi x = 4/3 (pi. 112/27).

7.3.2. y = x2

2 1, pi pi x = 0 (pi. 6 +

ln(

2 +

3)).

7.3.3. y = a cosh(xa) x = a, x = a (pi. 2a sinh(1)).

7.3.4. 9y2 = x(x 3)2 x (pi. 43).7.3.5. y2 = 4

9(2 x)3, pi x = 1 (pi. 28/3).

7.3.6. y = ln(1 x2), x = 1/2, x = 1/2 (pi. 2 ln(3) 1).7.3.7. y3 = 8x2, x = 1, x = 8 (pi. 1

27

(104

13 125)).7.3.8. y = x

4+36x

, x = 1, x = 2 ( pi. 17/12).

7.3.9. y = 23

(1 + x2)3/2, x = 0, x = 3 ( pi. 21).

7.3.10. y = x4+4824x

, x = 2, x = 4 ( pi. 17/6).

7.3.11. (7.2).

pi pi pipi pi .

7.3.12. pi pi y = x2, y = 0 x = 1. (pi. x = 3/4, y = 3/10).

7.3.13. pi pi y =a2 x2, y = 0. (pi. x = 0, y = 4a/3pi).

7.3.14. pi pi y = sin x, y = 0, x = 0 x = pi. (pi. x = pi/2,y = pi/8).

7.3.15. pi pi y = 1x, y = 0, x = 1 x = 2. (pi. x = 1

ln 2,

y = 14 ln 2

).

7.3.16. pi pi y = x2, y = x. (pi. x = 1/2, y = 4/10).

7.3.17. pi pi y = ex, y = ex x = 1. (pi. x = 2/(e 1)2,y = e (3e2 2 + e2)/(4 (e 1)2).7.3.18. pi pi pi y = 4 x2 pi. (pi. x = 3/4, y = 8/5).

7.3.19. pi pi y = x2, x = y2. (pi. x = 9/20, y = 9/20).

.

8

8.1

8.1.1. pi pi pi pi f (x), f (x) 0,x [a, b]. pi pi pi x ( ). , pi pi.

8.1: pi pi pi x.

8.1.2.

V = pi

ba

[f (x)]2 dx. (8.1)

8.1.3. pi pi pi

S = 2pi

ba

f (x)

1 +

(df

dx

)2dx. (8.2)

75

.

8. 76

8.1.4. , pi pi pi pi f (x),f (x) 0, x [a, b], pi pi y, pipi pi

S = 2pi

ba

x

1 +

(df

dx

)2dx (8.3)

.

8. 77

8.2

8.2.1. (8.1). 8.1. V pi pi

pi pi pi - [a, b] pi [x0, x1], [x1, x2], ...,[xN1, xN ], a = x0 < x1 < ... < xN xn xn1 = x = baN . pi Vn = pi [f (xn)]2 x

VN =Nn=1

pi [f (xn)]2 x. (8.4)

pi N pi. VN N

V

pi, (8.4) , pi

V = pi ba

[f (x)]2 dx.

pipi , pi pi pi (pi pi pi- ).

8.2.2. pi pi pi f (x) =sinx, x [0, pi], pi x. 8.2. (8.1)

8.2: pi

sinx, x [0, pi], pi x.

V = pi

pi0

(sinx

)2dx =

pi0

sinxdx = pi [cosx]x=pix=0 = 2pi.

.

8. 78

8.2.3. pi pi pi x2

a2+ y

2

b2= 1, pi x.

pi pi pipi pi pi y =b

1 x2a2

, x [a, a], pi x.

V = pi

aa

(b

1 x

2

a2

)2dx = pib2

[x x

3

3a2

]x=ax=a

=4piab2

3.

8.2.4. pi pi pi f (x) =x2 + 1, x [0, 1], pi x.

pi (8.2)

V = pi

10

(x2 + 1

)2dx = pi

[x5

5+

2x3

3+ x

]x=1x=0

=28pi

15.

8.2.5. pi pi pi pi x pi y = x2 x = y2.

8.3. V = V2 V1, pi V1

8.3: pi pi x pi y = x2 x = y2.

pi x = y2 V2 pi y = x2. pi, x = 0 x = 1.

V1 = pi

10

(x2)2dx =

pi

5

V2 = pi

10

(x1/2

)2dx =

pi

2

V = V2 V1 = pi10.

.

8. 79

8.2.6. pi pi pi pi x pi y = x2

2/a x2 + y2 = a2.

pi , V = V2V1,pi V1 pi y = x2

2/a V2 pi

x2 + y2 = a2. pi, x = a/2 x = a/2 .

V1 = 2pi

a/2a/2

(x2

2

a

)2dx =

1

5

2pia3,

V2 =4

3pia3 pi ,

V = V2 V1 = 43pia3

2

5a3.

8.2.7. pi pi pi f (x) = x2 + 1,x [0, 1] ( pi y) pi pi 8.4.

8.1 pi pipi. . V = V1 V2, pi

8.4: pi x2 + 1, x [0, 1], pi y.

V1 = pi 12 2 = 2pi V2 ( pi). V2 pi pi (8.1), x = g (y), .

y = x2 + 1 x =y 1

pi

V2 = pi

21

(y 1

)2dy = pi

[y2

2 2y

]y=2y=1

=pi

2.

.

8. 80

V = V1 V2 = 2pi pi2 = 3pi/2.8.2.8. pi pi pi pi pi pi f (x) = x3, x [0, 1], pi x.

(8.2)

S = 2pi

10

x3

1 + (3x2)2dx =2pi

36

10

(1 + 9x4

)1/2d(x4)

=pi

18

[(1 + 9x4)

3/2

3/2

]x=1x=0

=pi

27

(103/2 1) .

8.2.9. pi pi pi pi pi pi f (x) = x2, x [0,2], pi y.

pi pi y pi pi (8.2) . pi pi ( !) pi

S = 2pi

20

x

1 + (2x)2dx =

pi

8

20

(1 + 4x2

)1/2d(4x2)

=pi

4

[(1 + 4x2)

3/2

3/2

]x=2x=0

=13pi

3.

8.2.10. pi pi pi pi pi pi 9y2 = x (3 x)2, pi x.

pi x a = 0 b = 3. pi

y =

x (3 x)

3 dy

dx=

1 x2x.

pi

S = 2pi

30

x (3 x)

3

1 +

(1 x2x

)2dx = 2pi

30

16

(x+ 1) (x 3) dx

=pi

3

30

(2x+ 3 x2) dx = pi

3

[x2 + 3x x

3

3

]x=3x=0

= 3pi.

8.2.11. pi pi pi pi pi pi f (x) = ex, x 0, pi x.

8.5. f (x) x, pi . df

dx= ex

VM = 2pi

M0

ex

1 + (ex)2dx = 2pi x=Mx=0

(1 + e2x

)1/2d(ex)

= 2pi u=eMu=1

(1 + u2

)1/2du = 2pi

[u

1 + u2 + ln(u+

1 + u2)]u=1

u=eM.

.

8. 81

8.5: pi ex, x 0, pi x.

V = limM VM , pi

V = 2pi limM

[u

1 + u2 + ln(u+

1 + u2)]u=1

u=eM

= 2pi[u

1 + u2 + ln(u+

1 + u2)]u=1

u=0= 2pi

[2 + ln

(1 +

2)].

8.2.12. pi pi pi pi pi f (x) = coshx, x [1, 1], pi x.

E = 2pi

11

coshx

1 + sinh2 xdx.

1 + sinh2 x = cosh2 x pi

E = 2pi

11

cosh2 xdx = cosh 1 sinh 1 + 1.

.

8. 82

8.3

8.3.1. (8.2).

pi pi , pi.

8.3.2. pi pi pi pi f (x) =1/x+ 1, x [0, 3], pi x. (pi. pi ln 4)

8.3.3. pi pi pi pi f (x) =x

4 x2, pi x. (pi. 3pi/4)8.3.4. pi pi pi pi f (x) =ex, x [0, 1], pi x. (pi. pi

2 (1 e2))

8.3.5. pi pi pi pi f (x) =sinx, x [0, pi/2], pi x. (pi. pi)

8.3.6. pi pi pi pi f (x) =8x, x [0, 2], pi x. (pi. 16pi)8.3.7. pi pi pi pi f (x) =sinx, x [0, pi], pi x. (pi. pi2/2)8.3.8. pi pi pi pi f (x) =ex, x [0, 1], pi y. (pi. 2pi)8.3.9. pi pi pi pi f (x) =x3, x [0, 2], pi y. (pi. 19pi)8.3.10. pi pi pi pi pi f (x) = x3/3, x [0, 3], pi x. (pi. pi

9

(823/2 1))

8.3.11. pi pi pi pi pi f (x) = x

3

6+ 1

2x, x [1, 2], pi x. (pi. 47pi

16)

8.3.12. pi pi pi pi pi f (x) = x1/3 + 2, x [1, 8], pi y. (pi. pi

27

(1453/2 103/2))

8.3.13. pi pi pi pi pi f (x) = x

2

4 lnx

2, x [1, 4], pi x. (pi. 2pi

2

5(epi 2))

8.3.14. pi pi pi pi pi f (x) = x

2

4 lnx

2, x [1, e], pi x. (pi. pi

16(e4 9))

8.3.15. pi pi pi pi pi x2 + (y 2)2 = 1, pi x. (pi. 8pi2)

.

9 pi

pi pi pi y = f (x) pi. pi pi pi pi . pipi pi pi- pi pi pi : (x (t) , y (t)) t pi .

9.1

9.1.1. pi pi pi x(t), y(t), t pi [t1, t2]. , pi C

C : {(x (t) , y (t) , t [t1, t2])} .9.1.2. pi pi pi pi x(t), y(t) pi pi.9.1.3. pi t x(t), y(t) pi -pi. . pi pi t (x (t) , y (t)).9.1.4. pi pi pi pi pi pi.9.1.5. pi pi (x (t) , y (t)) tpi pi t1 t2, pi pi . pi pi

E =

t2t1

y (t)dx

dtdt =

t2t1

x (t)dy

dtdt.

9.1.6. pi pi (x (t) , y (t)) t pi pi t1 t2. pi

s =

t2t1

(dx

dt

)2+

(dy

dt

)2dt.

83

.

9. 84

9.1.7. pi pi pi pi y = f (x) pi pipi pi pidx = dx

dtdt, dy = dy

dtdt. pi 9.2.6, 9.2.7.

.

9. 85

9.2

9.2.1. pi x2 + y2 = R2. pi

x (t) = R cos t, y (t) = R sin t, t [0, 2pi] .

cos( t )

sin( t )

t

9.1: pi pi x (t) = R cos t, y (t) = R sin t, t [0, 2pi].

E =

2pi0

y (t)dx

dtdt =

2pi0

R sin t (R sin t) dt = R2 2pi

0

sin2 tdt

= R2 2pi

0

1 cos 2t2

dt =

[R2 t

2

2

]2pit=0

+

[R2

sin 2t

4

]2pit=0

= piR2 + 0 = piR2.

pipi pi . x2 + y2 = R2pi pipi pi. .. pipi

x (t) = R sin t, y (t) = R cos t, t [0, 2pi]

pi pi :

E =

2pi0

y (t)dx

dtdt =

2pi0

R cos t (R cos t) dt = R2 2pi

0

cos2 tdt = piR2.

9.2.2. pi x2a2

+ y2

b2= 1.

.

9. 86

pi x (t) = a cos t, y (t) = b sin t,t [0, 2pi].

E =

2pi0

y (t)dx

dtdt =

2pi0

b sin t (a sin t) dt = ab 2pi

0

sin2 tdt

= ab 2pi

0

1 cos 2t2

dt =

[abt

2

2

]2pit=0

+

[ab

sin 2t

4

]2pit=0

= piab.

9.2.3. pi pi x (t) = a cos3 t, y (t) = a sin3 t, t [0, 2pi] pi, pi 9.2.

9.2: pi pi x (t) = a cos3 t, y (t) = a sin3 t, t [0, 2pi].

pi

s =

t2t1

(dx

dt

)2+

(dy

dt

)dt =

2pi0

(3a cos2 t sin t)2 + (3a sin2 t cos t)2dt

= 3a

2pi0

cos4 t sin2 t+ sin4 t cos2 tdt = 3a

2pi0

cos2 t sin2 t (sin2 t+ cos2 t)dt

= 3a

2pi0

|cos t sin t| dt = 12a pi/2

0

sin (2t)

2dt = 6a.

9.2.4. pi pi x (t) = a (t sin t), y (t) = a (1 cos t),t [0, 2pi]

pi, pi pi 9.3.

.

9. 87

9.3: pi pi x (t) = a(t sin t), y (t) = a(1 cos t), t [0, 2pi].

pi

s =

t2t1

(dx

dt

)2+

(dy

dt

)dt =

2pi0

a2 (1 cos t)2 + a2 sin2 tdt

= a

2pi0

1 2 cos t+ cos2 t+ sin2 tdt = a

2pi0

2 (1 cos t)dt = a

2pi0

2 sint

2dt = 8a.

9.2.5. pi pi x (t) = t2, y (t) = t3, t [0, 4]

s =

40

(2t)2 + (3t2)2dt =

40

2t

1 +

3

2t2dt

=

t=4t=0

(1 +

3

2t2)1/2

d(t2)

=

u=16u=0

(1 +

3

2u

)1/2du =

8

27

(373/2 1) .

9.2.6. pi pi pi pi pi, pi x, x2 + y2 = 9 pi (3, 0)

(3/2, 3

3/2).

(3, 0) (3/2, 3

3/2) , pi pi

pi x2+y2 = 9. pi : x (t) = 3 cos t y (t) = 3 sin t, pi t t1 = 0 t2 = pi/3. pi pi pi pi x,

E = 2pi

x2x1

y

1 +

(dy

dx

)2dx = 2pi

t2t1

y (t)

(dx

dt

)2+

(dy

dt

)2dt.

.

9. 88

pi pi

E = 2pi

pi/30

3 sin t

(3 sin t)2 + (3 cos t)2dt = 2pi

pi/30

3 sin tdt = 9pi.

9.2.7. pi pi pi pi pi pi x(t) = a (t sin t), y(t) = a (1 cos t), t [0, 2pi] pi x.

pi 9.3. -

E = 2pi

t2t1

y (t)

(dx

dt

)2+

(dy

dt

)2dt

= 2pi

2pi0

a (1 cos t)a2 (1 cos t)2 + a2 sin2 t

= 2pia2 2pi

0

(1 cos t)

1 2 cos t+ cos2 t+ sin2 tdt

= 2pia2 2pi

0

(1 cos t)

2 (1 cos t)dt

= 4pia2 2pi

0

(1 cos t) sin t2dt =

64pia2

3.

.

9. 89

9.3

pi pi , pi.

9.3.1. pi pi pi pi x(t) = t sin t, y(t) = 1 cos t x. (pi. 3pi).

9.3.2. pi pi pi pi x(t) = cos3 t, y(t) = sin3 t.(pi. 3pi/8).

9.3.3. pi pi pi pi x(t) = a cos t, y(t) = b sin t.(pi. piab).

9.3.4. pi pi pi pi x(t) = 2 cos t cos 2t, y(t) =2 sin t sin 2t.(pi. 6pi).9.3.5. pi pi pi x(t) = 3t2, y(t) = 3t t3.(pi. 72

3/5).

9.3.6. pi pi pi x(t) = t21, y(t) = t3 t.(pi. 8/15).

9.3.7. pi x(t) = t sin t, y(t) = 1 cos t 0 t 2pi. (pi. 8).

9.3.8. pi pi x(t) = t6/6, y(t) = 2 t4/4 tx t ty,pi tx, ty pi pi x y. (pi. 13/3).

9.3.9. pi pi x(t) = t2, y(t) = t33 t t1 t t2, pi

t1, t2 pi pi x. (pi. 4/

3).

9.3.10. pi pi x(t) = et cos t, y(t) = et sin t 0 t pi.(pi.

2(epi 1)).

9.3.11. pi pi x(t) = 12

ln(1 + t2), y(t) = tan1 t 0 t 1. (pi. ln(2 + 1) ).9.3.12. pi pi x(t) = 2 cos(t) + cos(2t) + 1, y(t) = 2 sin t+sin 2t 0 t 2pi. (pi. 16 ).9.3.13. pi pi x(t) = t2

2, y(t) = 1

9(6t+ 9)3/2 0 t 4.

(pi. 20 ).

9.3.14. pi pi x(t) = cos3(t), y(t) = sin3(t) 0 t pi/2. (pi. 3/2 ).

9.3.15. pi pi x(t) = cos(t) + t sin(t), y(t) = sin t t cos t pi/6 t pi/4. (pi. 5pi2/288 ).9.3.16. pi pi x(t) = ln(sin(t)), y(t) = t pi/6 t pi/2.(pi. ln(2 +

3) ).

.

9. 90

9.3.17. pi pi pi pi pi x(t) = a (t sin t), y(t) = a (1 cos t), t [0, 2pi] pi x. (pi. 5pi2a3 ).

9.3.18. pi pi pi pi pi x(t) = a cos3 (t), y(t) = a cos3 (t),, t [0, 2pi] pi x.(pi. 32

105pia3 ).

9.3.19. pi pi pi pi pi pi x(t) = t, y(t) = 2t, t [0, 4] pi x. (pi. 32pi5 ).9.3.20. pi pi pi y.(pi. 16pi

5.)

9.3.21. pi pi pi pi pi pi x(t) = et sin t, y(t) = et cos t, t [0, pi/2] pi x. (pi.2pi

25

(epi 2).)9.3.22. pi pi pi pi pi pi x(t) = a cos3 (t), y(t) = a cos3 (t),, t [0, 2pi] pi x.(pi. 12

5pia2 ).

.

10

pi pi pipi (x, y). pi pi pi pi pi. pi , pi .

10.1

10.1.1. pipi pi (, ) . r pi Ox. pi pi (, ). 10.1.

cos( )

sin( )

(x,y)= (,)

10.1: .

91

.

10. 92

10.1.2. pi pi pi pi = (). pi () (, ()). 10.2.

,()

10.2: .

10.1.3. pi pi (x, y) pi (, r) :

x = cos, y = sin

=x2 + y2, = arctan

(yx

).

10.1.4. pi pi (, ()) pi pi 1 2, pi pi . pi pi

E =1

2

21

(())2 d. (10.1)

10.1.5. pi pi (, ()) pi pi 1 2. pi

s =

21

2 +

(d

d

)2d. (10.2)

.

10. 93

10.2

10.2.1. pi (x, y) = (1, 1). (1, 0). .

(x, y)=(1, 1) =

12 + 12 =

2, = arctan 11

= pi/4, .(, ) =

(2, pi/4

). (x, y)=(1, 0) =

12 + 02 =

1, = arctan 0

1= 0,

. (, ) = (1, 0).

10.2.2. pi (, ) = (2, pi/3). (1, 0). .

(, )=(2, pi/3) x = cos = 2 cos pi3

= 2 12

= 1, y = sin =2 sin pi

3= 2

3

2=

3, . (x, y) =(1,

3). (, )=(1, 0) x = cos =

1 cos 0 = 1, y = sin = 1 sin 0 = 0, . (x, y) = (1, 0).

10.2.3. (10.1). 10.3. pi E

10.3: pi pi .

[1, 2], EN N pi = 21

N, , N ,

EN . R A = 1

2R2. pi

E = limN

EN = limN

Nn=1

1

2( (n))

2 =1

2

21

2d.

10.2.4. (10.2). 10.4. pi s pi

[1, 2], sN N pi = 21

N, , N , -

sN . n- pi

.

10. 94

[()2+ ()2]1/2

10.4: pi pi .

pi pi (.pi ). pi

s = limN

sN = limN

Nn=1

( (n) )

2 + ()2

= limN

Nn=1

( (n))

2 +

(

)2 =

21

2 +

(d

d

)2d.

10.2.5. pi pi pi pi () =

2 cos 2, [pi/4, pi/4].

10.5.

10.5: pi pi .

.

10. 95

E =1

2

pi/4pi/4

2d =1

2

pi/2pi/2

2 cos2 2d =

pi/2pi/2

1 + cos 4

2d =

pi

2.

10.2.6. pi pi pi pi () = a (1 + cos), [pi, pi].

10.6. pi pi x, pi pi [0, pi] pi pi pi.

10.6: pi pi .

E = 2 a2

2

pi0

(1 + cos)2 d = a2 pi

0

(1 + 2 cos+ cos2

)d

= a2 ( pi

0

d+

pi0

2 cosd+

pi0

1 + cos 2

2d

)= a2

(pi + 0 +

pi

2

)=

3pia2

2.

.

10. 96

10.2.7. pi pi pi pi pi = sin 3. 10.7. pi , [0, pi/3]

pi pi pi.

10.7: pi pi .

E = 3 12

pi/30

(sin 3)2 d =3

2

pi/30

1 cos 62

d =pi

4.

10.2.8. pi pi () = e2 pi [0, 2pi].

10.8.

10.8: pi pi .

s =

2pi0

2 +

(d

d

)2d =

2pi0

e4 + 4e4d =

5

2

(e4pi 1) .

.

10. 97

10.2.9. pi pi R. R () = R, [0, 2pi] ( ;). pi

pi

s =

2pi0

2 +

(d

d

)2d =

2pi0

R2 + (0)2d =

2pi0

Rd = 2piR

pi pi.

10.2.10. pi pi () =a cos3

3.

10.9. = 0 = a. = 0 pipi cos3

3= 0, pi = 3pi

2+ 3kpi, k Z.

10.9: pi pi .

s = 2

3pi/20

a2 cos6

3+ a2 cos4

3sin2

3d = 2a

3pi/20

cos4

3(

cos2

3+ sin2

3

)d

= 2a

3pi/20

cos2

3d = 2a

3pi/20

1 + cos 23

2d =

3

2pia.

.

10. 98

10.2.11. pi pi () = 1/ pi [1/2, 2].

10.10.

10.10: pi pi .

s =

21/2

1

2+

( 12

)2d =

21/2

1

1 + 2

2d =

5

2+

1

2ln

(

5 1) (5 + 2)(5 + 1

) (5 2)

.10.2.12. pi pi pi pi pi pi( pi x) pi () = a (1 + cos), [pi, pi].

pi pi 10.6. - pi pi pi .

E = 2pi

ba

y

1 +

(dy

dx

)2dx = 2pi

ba

yds.

pi, pi pi

E = 2pi

21

sin

2 +

(d

d

)2d

pi, pi,

E = 2pi

pipia (1 + cos) sin

(a (1 + cos))2 + (a sin)2d

= 2pia2 pipi

(1 + cos) sin

2 (1 + cos)d

= 4pia2 pi

0

cos3

2sind = 8pia2

pi0

cos4

2sin

2d =

16

5pia2.

.

10. 99

10.11: pi pi .

10.2.13. pi pi pi pi pi pi( pi x) pi () = acos 2, [pi/4, pi/4] .

10.11. pi pi,

E = 4pi

pi/40

a

cos 2 sin a2 cos 2+ a2

sin2 2

cos 2d

= 4pia2 pi/4

0

cos 2 sin 1

cos 2d = 4pia2

pi/40

sind = 2pia2 (

2

2).

.

10. 100

10.3

( pi pi , pi.)

10.3.1. pi pi = , 0 2pi. (pi. 12

2pi02d =

12

2pi02d = 4

3pi3).

10.3.2. pi pi = e, 0 2pi. (pi. 12

2pi02d =

12

2pi0e2d = 1

4e4pi 1

4).

10.3.3. pi 2 = cos 2.(pi. 1).

10.3.4. pi = 1 cos 2. (pi. 3pi/2).10.3.5. pi = 3 + sin(2). (pi. 19pi/8).

10.3.6. pi = 2 cos(3). (pi. 3pi/4).10.3.7. pi = sin(2). (pi. pi/2).

10.3.8. pi = |cos(2)|. (pi. pi/2).10.3.9. pi = |sin(3)|. (pi. pi/4).10.3.10. pi = sin() + cos(). (pi. pi/2).

10.3.11. pi = 1/, pi4 2pi). (pi. E = 7/4pi).

10.3.12. pi 2 = 1 + sin.(pi. = 1 + sin. pi pi pi 4pi+2

4,

pi pi pi 4pi24

. 4pi+2

4 4 pi2

2= 4.)

10.3.13. pi = 1 cos. (pi. 8).10.3.14. pi pi pi = . (pi. pi 1 + 4pi2 ln(2pi +

1 + 4pi2

)).

10.3.15. pi = sin3 (/3). (pi. 3pi/2).

10.3.16. pi = 1/ pi 1 = 3/4 2 = 4/3. (pi. ln(

32

)+ 5

12).

10.3.17. pi = e pi 1 = 0 2 = ln 4. (pi. 3

2).

10.3.18. pi = 2 pi 1 = 0 2 =

5. (pi. 193).

.

11

pi N = {1, 2, 3, ...}. pi pi , pi .

11.1

11.1.1. f (n), pi n N = {1, 2, 3, ...} (. f : N R) . f (n) {fn}n=1 fn.11.1.2. {fn}n=1 pi A, B n : A fn B.11.1.3. {fn}n=1 () n : fn fn+1 (n :fn fn+1 ). , .11.1.4. {fn}n=1

> 0n : n n |fn | < . pipi fn limn fn = .

11.1.5. {fn}n=1 + ( ) M > 0nM : n nM fn > M (M < 0nM : n nM fn < M ). pipi fn + ( ) limn fn = + (limn fn = .).11.1.6. .

11.1.7. {fn}n=1 {gn}n=1 . limn fn = limn gn =. .

1. k R : limn (k fn) = k .2. limn (fn + gn) = + .

3. limn (fn gn) = .

101

.

11. 102

4. limn(fngn

)=

, 6= 0.

11.1.8. a |a| < 1 fn = an. limn fn= 0.

11.1.9. a a > 1 fn = an. limn fn=.11.1.10. fn. {sn}n=1 :

s1 = f1, s2 = f1 + f2, s3 = f1 + f2 + f3,..., sn = f1 + ...+ fn,...

s1, s2, s3, ... .

11.1.11. sN =N

n=1 fn limN sN pi

n=1 fn. , pi

n=1 fn pi.

11.1.12.

n=1 fn

n=1 fn < , pi ( ) n=1 fn = pipi.11.1.13. pi

1.

n=1 fn = a.

n=1 cfn = ca ( a pi pi).

2.

n=1 fn fn+1 ()lim fn = 0.

11.1.16.

n=1 fn pi

n=1 |fn|

.

11. 103

11.1.17. pi .

11.1.18.

n=1 fn = A R

n=1 gn = B R,

n=1 (fn + gn) = A+B.

11.1.19. a R,

1. |a| < 1Nn=0 an = 1 + a+ a2 + ... = 11a .2. a > 1Nn=0 an = 1 + a+ a2 + ... =.

.

11. 104

11.2

11.2.1. pi ( ) limn 1n = 0. > 0. n > 1 pi >

1n> 1

n n n. pi

n n : > 1n

=

1n > |fn 0| limn fn = 0.

11.2.2. pi ( ) limn(

1n2

+ 1)

= 1.

> 0. n >

1pi >

(1n

)2>(

1n

)2 n n.pi

n n : > 1n2

=

1n2 + 1 1 > |fn 1| limn fn = 1.

11.2.3. > 0n : n n |fn | < . . , n n, fn

n

fn

+

-

11.1: .

pi 2, pi . pi

n n |fn | < .

pi pi , pi (. pipi ). n pi pipi pi , . pi pi n.

.

11. 105

11.2.4. pi ( ) limn an = 0 |a| < 1. > 0. pi |a| < 1,

(|a| , 1). n > log|a| = ln ln|| > 0 ( ln < 0,ln |a| < 0).

n > log|a| n ln |a| < ln |a|n < |an 0| < n n > |an 0| > |an 0|, . limn fn = 0.11.2.5. pi limn 1n2+1 .

limn

1

n2 + 1= lim

n

1n2

n2

n2+ 1

n2

=limn 1n2

(limn 1) +(limn 1n2

) = 01 + 0

= 0.

11.2.6. pi limn n+1n2+1 .

limn

n+ 1

n2 + 1= lim

n

nn2

+ 1n2

n2

n2+ 1

n2

=

(limn 1n

)+(limn 1n2

)(limn 1) +

(limn 1n2

) = 0 + 01 + 0

= 0.

11.2.7. pi limn n2+n+1n2+1

.

limn

n2 + n+ 1

n2 + 1= lim

n

n2

n2+ n

n2+ 1

n2

n2

n2+ 1

n2

=limn 1 + limn 1n + limn

1n2

limn 1 + limn 1n2=

1 + 0 + 0

1 + 0= 1.

11.2.8. pi limn sin(n)n2+1 .

0 1. 1

1

xkdx =

[1