Problem Set: Integration by Parts

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals.

1. $\displaystyle\int {x}^{3}{e}^{2x}dx$

Show Solution

u = x^{3}

$u={x}^{3}$

\int x^{3} ln (x) d x

$\displaystyle\int {x}^{3}\text{ln}\left(x\right)dx$

3. $\displaystyle\int {y}^{3}\cos{y} dy$

Show Solution

u = y^{3}

$u={y}^{3}$

\int x^{2} arctan x d x

$\displaystyle\int {x}^{2}\text{arctan}xdx$

5. $\displaystyle\int {e}^{3x}\sin\left(2x\right)dx$

Show Solution

u = sin (2 x)

$u=\sin\left(2x\right)$

Find the integral by using the simplest method. Not all problems require integration by parts.

\int v sin v d v

$\displaystyle\int v\sin{v} dv$

7. $\displaystyle\int \text{ln}xdx$ (Hint: $\displaystyle\int \text{ln}xdx$ is equivalent to $\displaystyle\int 1\cdot \text{ln}\left( x\right)dx.$ )

Show Solution

- x + x ln x + C

$\text{-}x+x\text{ln}x+C$

\int x cos x d x

$\displaystyle\int x\cos{x}dx$

9. $\displaystyle\int {\tan}^{-1}xdx$

Show Solution

x {tan}^{- 1} x - \frac{1}{2} ln (1 + x^{2}) + C

$x{\tan}^{-1}x-\frac{1}{2}\text{ln}\left(1+{x}^{2}\right)+C$

10.

\int x^{2} e^{x} d x

$\displaystyle\int {x}^{2}{e}^{x}dx$

11. $\displaystyle\int x\sin\left(2x\right)dx$

Show Solution

- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x) + C

$-\frac{1}{2}x\cos\left(2x\right)+\frac{1}{4}\sin\left(2x\right)+C$

12.

\int x e^{4 x} d x

$\displaystyle\int x{e}^{4x}dx$

13. $\displaystyle\int x{e}^{\text{-}x}dx$

Show Solution

e^{- x} (- 1 - x) + C

${e}^{\text{-}x}\left(-1-x\right)+C$

14.

\int x cos 3 x d x

$\displaystyle\int x\cos3xdx$

15. $\displaystyle\int {x}^{2}\cos{x}dx$

Show Solution

2 x cos x + (- 2 + x^{2}) sin x + C

$2x\cos{x}+\left(-2+{x}^{2}\right)\sin{x}+C$

16.

\int x ln x d x

$\displaystyle\int x\text{ln}xdx$

17. $\displaystyle\int \text{ln}\left(2x+1\right)dx$

Show Solution

\frac{1}{2} (1 + 2 x) (- 1 + ln (1 + 2 x)) + C

$\frac{1}{2}\left(1+2x\right)\left(-1+\text{ln}\left(1+2x\right)\right)+C$

18.

\int x^{2} e^{4 x} d x

$\displaystyle\int {x}^{2}{e}^{4x}dx$

19. $\displaystyle\int {e}^{x}\sin{x}dx$

Show Solution

\frac{1}{2} e^{x} (- cos x + sin x) + C

$\frac{1}{2}{e}^{x}\left(\text{-}\cos{x}+\sin{x}\right)+C$

20.

\int e^{x} cos x d x

$\displaystyle\int {e}^{x}\cos{x}dx$

21. $\displaystyle\int x{e}^{\text{-}{x}^{2}}dx$

Show Solution

- \frac{e^{- x^{2}}}{2} + C

$-\frac{{e}^{\text{-}{x}^{2}}}{2}+C$

22.

\int x^{2} e^{- x} d x

$\displaystyle\int {x}^{2}{e}^{\text{-}x}dx$

23. $\displaystyle\int \sin\left(\text{ln}\left(2x\right)\right)dx$

Show Solution

- \frac{1}{2} x cos [ln (2 x)] + \frac{1}{2} x sin [ln (2 x)] + C

$-\frac{1}{2}x\cos\left[\text{ln}\left(2x\right)\right]+\frac{1}{2}x\sin\left[\text{ln}\left(2x\right)\right]+C$

24.

\int c o s (ln x) d x

$\displaystyle\int cos\left(\text{ln}x\right)dx$

25. $\displaystyle\int {\left(\text{ln}x\right)}^{2}dx$

Show Solution

2 x - 2 x ln x + x {(ln x)}^{2} + C

$2x - 2x\text{ln}x+x{\left(\text{ln}x\right)}^{2}+C$

26.

\int ln (x^{2}) d x

$\displaystyle\int \text{ln}\left({x}^{2}\right)dx$

27. $\displaystyle\int {x}^{2}\text{ln}xdx$

Show Solution

(- \frac{x^{3}}{9} + \frac{1}{3} x^{3} ln x) + C

$\left(\text{-}\frac{{x}^{3}}{9}+\frac{1}{3}{x}^{3}\text{ln}x\right)+C$

28.

\int {sin}^{- 1} x d x

$\displaystyle\int {\sin}^{-1}xdx$

29. $\displaystyle\int {\cos}^{-1}\left(2x\right)dx$

Show Solution

- \frac{1}{2} \sqrt{1 - 4 x^{2}} + x {cos}^{- 1} (2 x) + C

$-\frac{1}{2}\sqrt{1 - 4{x}^{2}}+x{\cos}^{-1}\left(2x\right)+C$

30.

\int x arctan x d x

$\displaystyle\int x\text{arctan}xdx$

31. $\displaystyle\int {x}^{2}\sin{x}dx$

Show Solution

- (- 2 + x^{2}) cos x + 2 x sin x + C

$\text{-}\left(-2+{x}^{2}\right)\cos{x}+2x\sin{x}+C$

32.

\int x^{3} cos x d x

$\displaystyle\int {x}^{3}\cos{x}dx$

33. $\displaystyle\int {x}^{3}\sin{x}dx$

Show Solution

- x (- 6 + x^{2}) cos x + 3 (- 2 + x^{2}) sin x + C

$\text{-}x\left(-6+{x}^{2}\right)\cos{x}+3\left(-2+{x}^{2}\right)\sin{x}+C$

34.

\int x^{3} e^{x} d x

$\displaystyle\int {x}^{3}{e}^{x}dx$

35. $\displaystyle\int x{\sec}^{-1}xdx$

Show Solution

\frac{1}{2} x (- \sqrt{1 - \frac{1}{x^{2}}} + x \cdot {sec}^{- 1} x) + C

$\frac{1}{2}x\left(\text{-}\sqrt{1-\frac{1}{{x}^{2}}}+x\cdot {\sec}^{-1}x\right)+C$

36.

\int x {sec}^{2} x d x

$\displaystyle\int x{\sec}^{2}xdx$

37. $\displaystyle\int x\text{cosh}xdx$

Show Solution

- cosh x + x sinh x + C

$\text{-}\text{cosh}x+x\text{sinh}x+C$

Compute the definite integrals. Use a graphing utility to confirm your answers.

38.

\int_{\frac{1}{e}}^{1} ln x d x

${\displaystyle\int }_{\frac{1}{e}}^{1}\text{ln}xdx$

39. ${\displaystyle\int }_{0}^{1}x{e}^{-2x}dx$ (Express the answer in exact form.)

Show Solution

\frac{1}{4} - \frac{3}{4 e^{2}}

$\frac{1}{4}-\frac{3}{4{\text{e}}^{2}}$

40.

\int_{0}^{1} e^{\sqrt{x}} d x (let u = \sqrt{x})

${\displaystyle\int }_{0}^{1}{e}^{\sqrt{x}}dx\left(\text{let}u=\sqrt{x}\right)$

41. ${\displaystyle\int }_{1}^{e}\text{ln}\left({x}^{2}\right)dx$

Show Answer

42.

\int_{0}^{π} x cos x d x

${\displaystyle\int }_{0}^{\pi }x\cos{x}dx$

43. ${\displaystyle\int }_{\text{-}\pi }^{\pi }x\sin{x}dx$ (Express the answer in exact form.)

Show Solution

2 π

$2\pi$

44.

\int_{0}^{3} ln (x^{2} + 1) d x

${\displaystyle\int }_{0}^{3}\text{ln}\left({x}^{2}+1\right)dx$ (Express the answer in exact form.)

45. ${\displaystyle\int }_{0}^{\frac{\pi}{2}}{x}^{2}\sin{x}dx$ (Express the answer in exact form.)

Show Solution

$-2+\pi$

46.

${\displaystyle\int }_{0}^{1}x{5}^{x}dx$ (Express the answer using five significant digits.)

47. Evaluate $\displaystyle\int \cos{x}\text{ln}\left(\sin{x}\right)dx$

Show Solution

$\text{-}\sin\left(x\right)+\text{ln}\left[\sin\left(x\right)\right]\sin{x}+C$

Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral.

48.

$\displaystyle\int {x}^{n}{e}^{x}dx={x}^{n}{e}^{x}-n\displaystyle\int {x}^{n - 1}{e}^{x}dx$

49. $\displaystyle\int {x}^{n}\cos{x}dx={x}^{n}\sin{x}-n\displaystyle\int {x}^{n - 1}\sin{x}dx$

Show Solution

50.

$\displaystyle\int {x}^{n}\sin{x}dx=\_\_\_\_\_\_$

51. Integrate $\displaystyle\int 2x\sqrt{2x - 3}dx$ using two methods:

Using parts, letting $dv=\sqrt{2x - 3}dx$
Substitution, letting $u=2x - 3$

Show Solution

$\frac{2}{5}\left(1+x\right){\left(-3+2x\right)}^{\frac{3}{2}}+C$ b.

$\frac{2}{5}\left(1+x\right){\left(-3+2x\right)}^{\frac{3}{2}}+C$

State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem.

52.

$\displaystyle\int x\text{ln}xdx$

53. $\displaystyle\int \frac{{\text{ln}}^{2}x}{x}dx$

Show Solution

Do not use integration by parts. Choose

$u$ to be

$\text{ln}x$ , and the integral is of the form

$\displaystyle\int {u}^{2}du$ .

54.

$\displaystyle\int x{e}^{x}dx$

55. $\displaystyle\int x{e}^{{x}^{2}-3}dx$

Show Solution

Do not use integration by parts. Let

$u={x}^{2}-3$ , and the integral can be put into the form

$\displaystyle\int {e}^{u}du$ .

56.

$\displaystyle\int {x}^{2}\sin{x}dx$

57. $\displaystyle\int {x}^{2}\sin\left(3{x}^{3}+2\right)dx$

Show Solution

Do not use integration by parts. Choose u to be

$u=3{x}^{3}+2$ and the integral can be put into the form

$\displaystyle\int \sin\left(u\right)du$ .

Sketch the region bounded above by the curve, the x-axis, and $x=1$ , and find the area of the region. Provide the exact form or round answers to the number of places indicated.

58.

$y=2x{e}^{\text{-}x}$ (Approximate answer to four decimal places.)

59. $y={e}^{\text{-}x}\sin\left(\pi x\right)$ (Approximate answer to five decimal places.)

Show Solution

Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated.

60.

$y=\sin{x},y=0,x=2\pi ,x=3\pi$ about the y-axis (Express the answer in exact form.)

61. $y={e}^{\text{-}x}$ $y=0,x=-1x=0$ ; about $x=1$ (Express the answer in exact form.)

Show Solution

$2\pi e$

62. A particle moving along a straight line has a velocity of

$v\left(t\right)={t}^{2}{e}^{\text{-}t}$ after t sec. How far does it travel in the first 2 sec? (Assume the units are in feet and express the answer in exact form.)

63. Find the area under the graph of $y={\sec}^{3}x$ from $x=0\text{to}x=1$ . (Round the answer to two significant digits.)

Show Solution

64. Find the area between

$y=\left(x - 2\right){e}^{x}$ and the x-axis from

$x=2$ to

$x=5$ . (Express the answer in exact form.)

65. Find the area of the region enclosed by the curve $y=x\cos{x}$ and the x-axis for

$\frac{11\pi }{2}\le x\le \frac{13\pi }{2}$ . (Express the answer in exact form.)

Show Solution

$12\pi$

66. Find the volume of the solid generated by revolving the region bounded by the curve

$y=\text{ln}x$ , the x-axis, and the vertical line

$x={e}^{2}$ about the x-axis. (Express the answer in exact form.)

67. Find the volume of the solid generated by revolving the region bounded by the curve $y=4\cos{x}$ and the x-axis, $\frac{\pi }{2}\le x\le \frac{3\pi }{2}$ , about the x-axis. (Express the answer in exact form.)

Show Solution

$8{\pi }^{2}$

68. Find the volume of the solid generated by revolving the region in the first quadrant bounded by

$y={e}^{x}$ and the x-axis, from

$x=0$ to

$x=\text{ln}\left(7\right)$ , about the y-axis. (Express the answer in exact form.)

Techniques of Integration: Supplemental Content

Problem Set: Integration by Parts

Candela Citations