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• Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000www.tekoclasses.com Question. & Solution. Int. Cal. Page: 1 of 26

THE BOND || Phy. by Chitranjan|| ||Chem. by Pavan Gubrele|| ||Maths by Suhaag Kariya||

TOPIC = INTEGRAL CALCULUS

Single Correct Type

Que. 1. Let 3 3 32 4

f (x) sin x sin x sin x3 3

= + + + +

then the primitive of f(x) w.r.t. x is

(a) 3sin 3x

C4

+ (b) 3cos3x

C4

+ (c) sin 3x

C4

+ (d) cos3x

C4

+

where C is an arbitrary constant. (code-V2T3PAQ5)

Que. 2. If the dependent variable y is changed to z by the substitution y = tan z then the differential equa-

tion

22

2 2

d y 2(1 y) dy1

dx 1 y dx

+ = +

+ is changed to

2

2

d z

dx =

2

2 dzcos z k ,dx

+

then the value of k equals

(a) 1 (b) 0 (c)1 (d) 2 (code-V2T3PAQ10)

Que. 3. ( )xx n ex dx is equal to (code-V2T5PAQ1)

(a) xx C+ (b) x. n x (c) ( )

xc n x C+ (d) None

Que. 4. The value of the definite integral 2008

x

2008

f '(x) f '( x)dx

(2008) 1

+

+ equals (code-V2T5PAQ3)

(a) f (2008) f ( 2008)+ (b) f (2008) f ( 2008)

(c) 0 (d) f ( 2008) f (2008)

Que. 5.e

1

1 n xdx

xx n x

+

equals (code-V2T5PAQ4)

(a) e (b) 2e (c) 2 e (d) 2e

Que. 6. If x

4

0

g(x) cos t dt,= then g(x )+ equals (code-V2T5PAQ5)

(a) g(x) g( )+ (b) g(x) g( ) (c) g(x)g( ) (d) [ ]g(x) g( )

Que. 7. Let f be a positive function. Let ( ) ( )k k

1 2

1 k 1 k

I x f x(1 x) dx; I f x(1 x) dx,

= = where 2k 1 0. >

Then 2

1

I

I is (code-V2T5PAQ7)

(a) k (b) 1/2 (c) 1 (d) 2

Que. 8. 2 2

dx

x 16 x has the value equal to (code-V2T5PAQ8)

(a) 1 x

C arc sec4 4

(b)

1 xarc sec C

4 4

+

(c) 2

16 xC

16x

(d)

216 x

C16x

+

• Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000www.tekoclasses.com Question. & Solution. Int. Cal. Page: 2 of 26

THE BOND || Phy. by Chitranjan|| ||Chem. by Pavan Gubrele|| ||Maths by Suhaag Kariya||

Que. 9. If ( )2 2x 2x 2x .e dx e ax bx c d, = + + + then (code-V2T5PAQ10)

(a) 1 1 1

a ,b ,c2 2 4

= = = (b) 1 1 1

a ,b ,c2 2 4

= = =

(c) 1 1

a ,b 1,c2 2

= = = (d) 1

a 1,b 1,c2

= = =

Que. 10. If

1

-1

0

1tan x dx= - ln 2

4 2 then the value of the definite integral ( )1

1 2

0

tan 1 x x dx

+ equals

(a) n 2 (b) n 24

+ (c) n 2

4

(d) 2 n 2 (code-V2T5PAQ11)

Que. 11.

2

n

nn0

nlim x dx

2 equals (code-V2T5PAQ12)

(a) 1 (b) 2 (c) 1

2(d) 4

Que. 12. If f is continuous function and x 2

0 t

F(x) (2t 3). f (u)du dt

= + then F"(2) is equal to

(a) 7f (2) (b) 7f '(2) (c) 3f '(2) (d) 7f (2) (code-V2T5PAQ15)

Que. 13. ( )( )sin x/ 2

x 1 x cos x n x sin x dx

+ + is equal to (code-V2T5PAQ16)

(a)

2

2

(b)

2

(c)

24

4

(d) 1

2

Que. 14. Let f :[0, ) R be a continuous strictly increasing function, such that (code-V2T5PAQ17)

x

3 2

0

f (x) t.f (t)dt= for every x 0. The value of f (6) is

(a) 1 (b) 6 (c) 12 (d) 36

Que. 15. If the value of definite integral

/ 4

x

/ 6

1 cot xdx,

e sin x

+ is equal to / 6 / 4ae be + then (a + b) equals

(a) 2 2 (b) 2 2+ (c) 2 2 2 (d) 2 3 2 (code-V2T5PAQ18)

Que. 16. Let 30

n xJ dx

1 x

=+

and 20

x n xK dx

1 x

=+

then (code-V2T5PAQ20)

(a) J K 0+ = (b) J K 0 = (c) J K 0+ < (d) none

• Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000www.tekoclasses.com Question. & Solution. Int. Cal. Page: 3 of 26

THE BOND || Phy. by Chitranjan|| ||Chem. by Pavan Gubrele|| ||Maths by Suhaag Kariya||

Que. 17. The value of x > 1 satisfying the equation x

1

1t n t dt ,

4= is (code-V2T5PAQ22)

(a) e (b) e (c) 2e (d) e 1

Que. 18. If

x

1

F(x) f (t)dt= where 2

t 4

1

1 uf(t) du

u

+= then the value of F"(2) equals (code-V2T5PAQ23)

(a) 7

4 17(b)

15

17(c) 257 (d)

15 17

68

Que. 19. Let f be a continuous function on [a, b]. If ( )( )x b

a x

F(x) f (t)dt f (t)dt 2x a b

= + then there exist

some ( )c a,b such that (code-V2T8PAQ6)

(a)

c b

a c

f (t)dt f (t)dt= (b) ( )c b

a c

f (t)dt f (t)dt f (c) a b 2c = +

(c) ( )( )c b

a c

f (t)dt f (t)dt f (c) 2c a b = + (d) ( )( )c b

a c

f (t)dt f (t)dt f (c) 2c a b+ = +

Que. 20. The value of the definite integral ( ) ( )/ 2

x 2 2

0

x xI e cos sin x cos sin sin x sin dx,

2 2

= +

is

(a) ( )/ 21 e cos1 sin1 1

2

+ (b) ( )/ 2e

cos1 sin12

+ (codeV2T10PAQ2)

(c) ( )/ 21

e cos1 12

(d) [ ]

/ 2ecos1 sin1 1

2

+

(a) ( )/ 21 e cos1 sin1 1

2

+ (b) ( )/ 2e

cos1 sin12

+

(c) ( )/ 21

e cos1 12

(d) [ ]

/ 2ecos1 sin1 1

2

+

Que. 21. A tank with a capacity of 1000 liters originally contains 100 gms of salt dissolved in 400 liters of

water. Beginning at time t 0= and ending at time t 100= minutes, water containing 1 gm of salt per

liters enters the tank at the of 4 liters per minute, and the well mixed solution is drained from the tank

at a rate of 2 liter/minute. The differential equation for the amount of salt y in the tank at time t is

(a) dy y

4dt 400 2t

= +

(b) dy y

4dt 200 t

= +

(code-V2T11PAQ2)

(c) ( )

dy y4

dt 200 t 2=

+ (d) dy y

4dt 500 t

= +

• Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000www.tekoclasses.com Question. & Solution. Int. Cal. Page: 4 of 26

THE BOND || Phy. by Chitranjan|| ||Chem. by Pavan Gubrele|| ||Maths by Suhaag Kariya||

Que. 22. Let y y(t)= be solution to the differential equation 2y ' 2ty t ,+ = then t

ylim

t is (code-V2T11PAQ4)

(a) zero (b) 1

2(c) 1 (d) Non existent.

Que. 23. The area of the region bounded below by 1y sin x,= above by 1y cos x= and on the left by y-axis,

is

(a) 2 1 (b) 2 2 (c) 2 1+ (d) 2 (code-V2T11PAQ7)

Que. 24. 2

0

xdx

4 x is equal to (code-V2T13PAQ20)

(a) 2

(b) 1

2

(c) 1 (d) 2

Que. 25. If 2

4 2

2

x . 4 x dx

has the value equal to k then the value of k equals (code-V2T14PAQ20)

(a) 0 (b) 2 (c) 8 (d) 4

Que. 26. Let q

2008 r

dx 1 xn C

x x p 1 x

= +

+ + where p,q.r N and need not be distinct, then the value of

( )p q r+ + equals (code-V2T14PAQ25)

(a) 6024 (b) 6022 (c) 6021 (d) 6020

Que. 27. ( ) ( )/ 2

x

0

sin x n(sin x) x cot x dx

+ is (code-V2T17PAQ7)

(a) 1 (b) 1 (c) 0 (d) Indeterminant

Que. 28. Let ( )2

y n 1 cos x= + then the value of 2

2 y / 2

d y 2

dx e+ equals (code-V2T17PAQ8)

(a) 0 (b) 2

1 cos x+(c)

4

(1 cos x)+(d) 2

4

(1 cos x)

+

Que. 29. The value of definite integral 1

x 2

1

dx

(1 e )(1 x )

+ + is (code-V2T18PAQ1)

(a) / 2 (b) / 4 (c) / 8 (d) /16

Que. 30. The expression 2

3

2

d yy

dx on the ellipse 2 23x 4y 12+ = is equal to (code-V2T19PAQ5)

(a) 9

4(b)

9

4 (c)

4

9(d)

4

9

Que. 31. The value of the definite integral ( )n

0

x sin xdx n N

1 cos x

+

is equal (code-V2T19PAQ6)

(a) 2n n 2 (b) 2n n 2 (c) n n 2 (d) 2

n n 2

2

• Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000www.tekoclasses.com Question. & Solution. Int. Cal. Page: 5 of 26

THE BOND || Phy. by Chitranjan|| ||Chem. by Pavan Gubrele|| ||Maths by Suhaag Kariya||

Comprehesion Type# 1 Paragraph for Q. 1 to Q. 3

A curve in represented parametrically by the equations t tx e cos t and y e sin t= = is a parameter. Then

1. The relation between the parameter t and the angle between the tangent to the given curve and the

x-axis is given by, t equals (code-V2T4PAQ1,2,3)

(a) 2

(b)

4

+ (c)

4

(d)

4

2. The value of 2

2

d y

dx at the point where t = 0 is

(a) 1 (b) 2 (c) 2 (d) 3

3. If ( )F(t) x y dt= + the point the value of F F(0)2

is

(a) 1 (b) 1 (c) / 2e (d) 0

# 2 Paragraph for Q. 4 to Q. 6

Let f (x) is a derivable function satisfying ( )x

2

0

xf (x) f (t) dt x n x 1 x = + + with f (0) n 2.= Let

g(x) xf '(x)= then (code-V2T4PAQ4,5,6,)

4. Range of g (x) is

(a) [ )0, (b) [ )0,1 (c) [ )1, (d) [ ),

5. For the function f which one of the following is correct ?

(a) f is neither odd nor even (b) f is transcendental

(c) f is injective (d) f is symmetric w.r.t. origin.

6.

1

0

f (x) dx equals

(a)