Exercise4 - t Umathstat.sci.tu.ac.th/~archara/MA111/MA111-116/exercise...MA111:...

MA111: Prepared by Asst.Prof.Dr.Archara Pacheenburawana 1

Exercise 4.1

1. Find the most general antiderivative of the function.

(a) f(x) = 5 (b) f(x) = x2 + π

(c) f(x) = x5/4 (d) f(x) = 1/3√x2

(e) f(x) = x2 − x (f) f(x) = 4x5 − x3

(g) f(x) = 27x7 + 3x5 − 45x3 +√2x

(h) f(x) =3

x2− 2

x3(i) f(x) =

4x6 + 3x4

x3

2. Evaluate the integral and check your answer by differentiating.

(a)

∫

3x4 dx (b)

∫

(x2 + x) dx

(c)

∫

(3x4 − 3x) dx (d)

∫

(x+ 1)2 dx

(e)

∫

3√x dx (f)

∫(

3− 1

x4

)

dx

(g)

∫

x1/3 − 3

x2/3dx (h)

∫

(x2 + 1)2√x

dx

(i)

∫

(sin x− cosx) dx (j)

∫

2 sec x tan x dx

(k)

∫

5 sec2 x dx (l)

∫

(3ex − 2) dx

(m)

∫

(3 cosx− 1/x) dx (n)

∫(

5x− 3

ex

)

dx

(o)

∫

ex + 3

exdx (p)

∫

x1/4(x5/4 − 4) dx

Answer to Exercise 4.1

1. (a) 5x+ C (b) 1

3x3 + πx+ C (c) 4

9x9/4 + C (d) 3 3

√x+ C (e) 1

3x3 − 1

2x2 + C

(f) 2

3x6 − 1

4x4 + C (g) 27

8x8 + 1

2x6 − 45

4x4 +

√2

2x2 + C (h) − 3

x+ 1

x2 + C

(i) x4 + 3

2x2 + C

2. (a) 3

5x5 + C (b) 1

3x3 + x2

2+ C (c) 3

5x5 − 3

2x2 + C (d) 1

3(x+ 1)3 + C

(e) 2x3/2+C (f) 3x+ 1

3x−3+C (g) 3

2x2/3−9x1/3+C (h) 2

9x9/2+ 4

5x5/2+2x1/2+C

(i) − cos x− sin x+ C (j) 2 sec x+ C (k) 5 tanx+ C (l) 3ex − 2x+ C

(m) 3 sinx− ln |x|+C (n) 5

2x2 + 3e−x +C (o) x− 3e−x +C (p) 2

5x5/2 − 16

5x5/4


Exercise 4.2

1. Evaluate the integral by making the given substitution.

(a)

∫

cos 3x dx, u = 3x

(b)

∫

x2(x3 + 2) dx, u = x3 + 2

(c)

∫

4

(1 + 2x)3dx, u = 1 + 2x

(d)

∫

(√x+ 2)3√

xdx, u =

√x+ 2

2. Evaluate the indefinite integral

(a)

∫

2x(x2 + 3)4 dx (b)

∫

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

(c)

∫ √x− 1 dx (d)

∫

x2

√x3 − 2

dx

(e)

∫

dx

5− 3x(f)

∫

2x+ 1

x2 + x− 1dx

(g)

∫

1 + 4x√1 + x+ 2x2

dx (h)

∫

1√x(√x+ 1)2

dx

(i)

∫

cos 2x dx (j)

∫

cosx√sin x+ 1 dx

(k)

∫

x sin(x2) dx (l)

∫

sin x√cosx

dx

(m)

∫

cos4 x sin x dx (n)

∫

sin x(cos x+ 3)3/4 dx

(o)

∫

sec x tanx√1 + sec x dx (p)

∫

cosxesinx dx

(q)

∫

ex√1 + ex dx (r)

∫

xex2+1 dx

(s)

∫

dx

x ln x(t)

∫

4

x(ln x+ 1)2dx

(u)

∫ √cot x csc2 x dx (v)

∫

sin x(cosx− 1)3 dx

(w)

∫

sec3 x tanx dx (x)

∫

ex − e−x

ex + e−xdx

(y)

∫

1 + x

1 + x2dx (z)

∫

2x+ 3

x+ 7dx



1. (a) 1

3sin 3x+C (b) 2

3(x3 + 2)3/2 +C (c) −1/(1 + 2x)2 +C (d) 1

2(√x+ 2)4 +C

2. (a) 1

5(x2 + 3)5 + C (b) 1

4(x2 + x)4 + C (c) 2

3(x− 1)3/2 + C (d) 2

3

√x3 − 2 + C

(e) −1

3ln |5− 3x|+ C (f) ln |x2 + x− 1|+ C (g) 2

√1 + x+ 2x2 + C

(h) −2(√x+1)−1+C (i) 1

2sin 2x+C (j) 2

3(sin x+1)3/2+C (k) −1

2cos(x2)+C

(l) −2√cosx+ C (m) −1

5cos5 x+ C (n) −4

7(cosx+ 3)7/4 + C

(o) 2

3(1 + sec x)3/2 + C (p) esinx + C (q) 2

3(1 + ex)3/2 + C (r) 1

2ex

2+1 + C

(s) ln | lnx|+C (t) −4(ln x+1)−1+C (u) −2

3(cotx)3/2+C (v) −1

4(cosx−1)4+C

(w) 1

3sec3 x+ C (x) ln(ex + e−x) + C (y) tan−1 x+ 1

2ln(1 + x2) + C

(z) 2(x+ 7)− 11 ln |x+ 7|+ C

Exercise 4.3

1. Express the limit as a definite integral on the given interval.

(a) limn→∞

n∑

i=1

xi sin xi4x, [0, π]

(b) limn→∞

n∑

i=1

exi

1 + xi4x, [1, 5]

(c) limn→∞

n∑

i=1

[2(x∗i )

2 − 5x∗i ]4x, [0, 1]

(d) limn→∞

n∑

i=1

√

x∗i 4x, [1, 4]

2. Use the form of the definition of the integral given in (4.7) to evaluate the integral.

(a)

∫

1

0

2x dx (b)

∫

5

−1

(1 + 3x) dx

(c)

∫ 5

1

(2 + 3x− x2) dx (d)

∫ 2

0

(2− x2) dx

(e)

∫ 5

0

(1 + 2x3) dx (f)

∫ 2

1

x3 dx

3. Prove that

∫ b

a

x dx =b2 − a2

2.

4. Prove that

∫ b

a

x2 dx =b3 − a3

3.


5. Write the given sum or difference as a single integral in the form∫ b

af(x) dx.

(a)

∫ 2

0

f(x) dx+

∫ 3

2

f(x) dx (b)

∫ 3

0

f(x) dx−∫ 3

2

f(x) dx

(c)

∫

2

0

f(x) dx+

∫

1

2

f(x) dx (d)

∫

2

−1

f(x) dx+

∫

3

2

f(x) dx

(e)

∫

3

1

f(x) dx+

∫

6

3

f(x) dx+

∫

12

6

f(x) dx

6. If

∫ 8

2

f(x) dx = 1.8 and

∫ 8

5

f(x) dx = 3.2, find

∫ 5

2

f(x) dx.

7. If

∫ 1

0

f(x) dx = 3,,

∫ 4

0

f(x) dx = −7, and

∫ 4

3

f(x) dx = 2, find

∫ 3

1

f(x) dx.


1. (a)

∫ π

0

x sin xdx (b)

∫

5

1

ex

1 + xdx (c)

∫

1

0

(2x2 − 5x)dx (d)

∫

4

1

√x dx

2. (a) 1 (b) 42 (c) 2 (d) 4

3(e) 317.5 (f) 3.75

5. (a)

∫

3

0

f(x) dx (b)

∫

2

0

f(x) dx (c)

∫

1

0

f(x) dx (d)

∫

3

−1

f(x) dx (e)

∫

12

1

f(x) dx

6. −1.4 7. −12

Exercise 4.4

1. Use the First Fundamental Theorem of Calculus to find the derivative.

(a) f(x) =

∫ x

0

(t2 − 3t+ 2) dt (b) g(x) =

∫ x

0

√1− 2t dt

(c) g(y) =

∫ y

2

t2 sin t dt (d) F (x) =

∫ 2

x

cos(t2) dt

(e) f(x) =

∫ x2

0

(e−t2 + 1) dt (f) f(x) =

∫ −1

x

ln(t2 + 1) dt

(g) h(x) =

∫ 1/x

2

arctan t dt (h) y =

∫

√x

3

cos t

tdt

(i) y =

∫ 1

1−3x

u3

1 + u2du (j) y =

∫ x3

√x

√t sin t dt


2. Evaluate the definite integrals using the Second Fundamental Theorem of Calculus.

(a)

∫

3

−1

x5 dx (b)

∫

2

0

(2x− 3) dx

(c)

∫ 4

0

√x dx (d)

∫ 4

0

(√x+ 3x) dx

(e)

∫ 2

1

3

x4dx (f)

∫ 1

0

(x√x+ x−1/2) dx

(g)

∫ 3

3

√x5 + 2 dx (h)

∫ π/2

0

2 sin x dx

(i)

∫ π

π/2

sec x tan x dx (j)

∫ π

π/2

(2 sin x− cosx) dx

(k)

∫

9

1

1

2xdx (l)

∫

1

0

(ex − e−x) dx

(m)

∫ 9

8

2x dx (n)

∫ 3

0

(3ex − x2) dx

(o)

∫ π/2

π/6

(

x+2

sin2 x

)

dx (p)

∫

√3

1

6

1 + x2dx

(q)

∫ 2

0

|2x− 3| dx (r)

∫ 3π/4

0

| cosx| dx

(s)

∫

3

−2

f(x) dx, where f(x) =

{

−x, x ≥ 0x2, x < 0

(t)

∫

2

0

f(x) dx, where f(x) =

{

x4, 0 ≤ x < 1x5, 1 ≤ x ≤ 2


1. (a) f ′(x) = x2 − 3x+ 2 (b) g′(x) =√1 + 2x (c) g′(y) = y2 sin y

(d) F ′(x) = − cos(x2) (e) f ′(x) = (e−x4

+ 1)2x (f) f ′(x) = − ln(x2 + 1)

(g) h′(x) = − arctan(1/x)/x2 (h) y′ =cos

√x

2x(i) y′ =

3(1− 3x)3

1 + (1− 3x)2

(j) y′ = 3x7/2 sin(x3)− (sin√x)/(2 4

√x)

2. (a) 364

3(b) −2 (c) 16

3(d) 88

3(e) 7

8(f) 12

5(g) 0 (h) 2 (i) Does not exist

(j) 3 (k) ln 3 (l) e+ e−1 − 2 (m) 28

ln 2(n) 3e3 − 12 (o) π2

9+ 2

√3 (p) π

2

(q) 5

2(r) 2−

√2

2(s) −11

6(t) 10.7


Exercise 4.5

Evaluate the definite integral, if it exists.

1.

∫ 2

0

(x− 1)25 dx 2.

∫ 2

0

x√x2 + 1 dx

3.

∫ 1

0

x2(1 + 2x3)5 dx 4.

∫ 1

−1

x

(x2 + 1)2dx

5.

∫

1

0

cosπx dx 6.

∫ π

π/2

4 cosx

(sin x+ 1)2dx

7.

∫ 4

1

1

x2

√

1 +1

xdx 8.

∫ π/2

π/4

cotx dx

9.

∫

3

0

dx

2x+ 310.

∫

4

1

x− 1√x

dx

11.

∫ π/3

0

sin x

cos2 xdx 12.

∫ 13

0

dx3

√

(1 + 2x)2

13.

∫ 2

1

x√x− 1 dx 14.

∫ e4

e

dx

x√lnx

15.

∫ a

0

x√x2 + a2 dx (a > 0)


1. 0 2. 5

3

√5− 1

33. 81

44. 0 5. 0 6. −2 7. 4

√2

3− 5

√5

128. 1

2ln 2 9. 1

2ln 3

10. 8

311. 1 12. 3 13. 16

1514. 2 15. 1

3(2√2− 1)a3

Extra Problems

1. Use Part I of the Fundamental Theorem of Calculus to find the derivative of the

function y =

∫ 0

√x

(

1 + sin3 t)

dt.

2. Evaluate the following integrals. Explain your answer.

(a)

∫ π/2

0

d

dx

(

sinx

2cos

x

3

)

dx

(b)d

dx

∫ π/2

0

(

sinx

2cos

x

3

)

dx

(c)d

dx

∫ π/2

x

(

sint

2cos

t

3

)

dt

3. Let F (t) =

∫ t2

1

√1 + u4

udu, find F ′′(1).


4. Use the Fundamental Theorem of Calculus to find the derivative of the function

F (x) =

∫ 3

tan x

√

t+√t dt.

Answer F ′(x) = −√

tan x+√tan x · sec2 x

5. Find the derivative of the function g(x) =

∫ x3

x2

t2√1 + t dt.

Answer g′(x) = −2x5√1 + x2 + 3x8

√1 + x3

6. Use the Fundamental Theorem of Calculus to find the derivative of the function

F (x) =

∫ x3

√x

ln(t+ 1)

tdt.

Answer F ′(x) = − ln (√x+ 1)

2x+

3 ln (x3 + 1)

x

7. Use the Fundamental Theorem of Calculus to find the derivative of the functionF (x) when

F (x) =

∫ x2

√x

cos t

t2 + 3t+ 5dt.

Answer F ′(x) = − cos√x

2√x(x+ 3

√x+ 5)

+2x cosx2

x4 + 3x2 + 5

8. Use the Fundamental Theorem of Calculus to find the derivative of the functionF (x) when

F (x) =

∫ 0

3+tan x

sin(t2) dt.

Answer F ′(x) = − sin(3 + tanx)2 sec2 x

9. Given f(x) =

∫ x

0

1

1 + t + t2dt. Find f ′′(0).

10. (a) Evaluate

∫[

2−1/x

x2+ x sin(x2) cos(x2)

]

dx.

(b) Find the derivative of the function g(x) =

∫

4

lnx

tanh t dt.

Answer (a)2−1/x

ln 2+

sin2(x2)

4+ C (b) −tanh(ln x)

x

Exercise4 - t Umathstat.sci.tu.ac.th/~archara/MA111/MA111-116/exercise...MA111:...

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