Integration by Parts

In the following exercises (1-3), use the guidelines in this section to choose u. Do not evaluate the integrals.

1. [latex]\displaystyle\int {x}^{3}{e}^{2x}dx[/latex]
2.  [latex]\displaystyle\int {y}^{3}\cos{y} dy[/latex]
3. [latex]\displaystyle\int {e}^{3x}\sin\left(2x\right)dx[/latex]

For the following exercises (4-19), find the integral by using the simplest method. Not all problems require integration by parts.

4. [latex]\displaystyle\int \text{ln}xdx[/latex] (Hint: [latex]\displaystyle\int \text{ln}xdx[/latex] is equivalent to [latex]\displaystyle\int 1\cdot \text{ln}\left( x\right)dx.[/latex])
5. [latex]\displaystyle\int {\tan}^{-1}xdx[/latex]
6.  [latex]\displaystyle\int x\sin\left(2x\right)dx[/latex]
7. [latex]\displaystyle\int x{e}^{\text{-}x}dx[/latex]
8. [latex]\displaystyle\int {x}^{2}\cos{x}dx[/latex]
9.  [latex]\displaystyle\int \text{ln}\left(2x+1\right)dx[/latex]
10. [latex]\displaystyle\int {e}^{x}\sin{x}dx[/latex]
11. [latex]\displaystyle\int x{e}^{\text{-}{x}^{2}}dx[/latex]
12.  [latex]\displaystyle\int \sin\left(\text{ln}\left(2x\right)\right)dx[/latex]
13. [latex]\displaystyle\int {\left(\text{ln}x\right)}^{2}dx[/latex]
14.  [latex]\displaystyle\int {x}^{2}\text{ln}xdx[/latex]
15.  [latex]\displaystyle\int {\cos}^{-1}\left(2x\right)dx[/latex]
16. [latex]\displaystyle\int {x}^{2}\sin{x}dx[/latex]
17. [latex]\displaystyle\int {x}^{3}\sin{x}dx[/latex]
18. [latex]\displaystyle\int x{\sec}^{-1}xdx[/latex]
19. [latex]\displaystyle\int x\text{cosh}xdx[/latex]

For the following exercises (20-24), compute the definite integrals. Use a graphing utility to confirm your answers.

20.  [latex]{\displaystyle\int }_{0}^{1}x{e}^{-2x}dx[/latex] (Express the answer in exact form.)
21. [latex]{\displaystyle\int }_{1}^{e}\text{ln}\left({x}^{2}\right)dx[/latex]
22. [latex]{\displaystyle\int }_{\text{-}\pi }^{\pi }x\sin{x}dx[/latex] (Express the answer in exact form.)
23.  [latex]{\displaystyle\int }_{0}^{\frac{\pi}{2}}{x}^{2}\sin{x}dx[/latex] (Express the answer in exact form.)
24. Evaluate [latex]\displaystyle\int \cos{x}\text{ln}\left(\sin{x}\right)dx[/latex]

Derive the following formulas (25-26) using the technique of integration by parts. Assume that [latex]n[/latex] is a positive integer. These formulas are called reduction formulas because the exponent in the [latex]x[/latex] term has been reduced by one in each case. The second integral is simpler than the original integral.

25. [latex]\displaystyle\int {x}^{n}\cos{x}dx={x}^{n}\sin{x}-n\displaystyle\int {x}^{n - 1}\sin{x}dx[/latex]

26. Integrate [latex]\displaystyle\int 2x\sqrt{2x - 3}dx[/latex] using two methods:

    1. Using parts, letting [latex]dv=\sqrt{2x - 3}dx[/latex]
    2. Substitution, letting [latex]u=2x - 3[/latex]

In the following exercises (27-29), state whether you would use integration by parts to evaluate the integral. If so, identify [latex]u[/latex]and [latex]dv.[/latex] If not, describe the technique used to perform the integration without actually doing the problem.

27. [latex]\displaystyle\int \frac{{\text{ln}}^{2}x}{x}dx[/latex]
28.  [latex]\displaystyle\int x{e}^{{x}^{2}-3}dx[/latex]
29.  [latex]\displaystyle\int {x}^{2}\sin\left(3{x}^{3}+2\right)dx[/latex]

In the following exercise, sketch the region bounded above by the curve, the [latex]x[/latex]-axis, and [latex]x=1[/latex], and find the area of the region. Provide the exact form or round answers to the number of places indicated.

30. [latex]y={e}^{\text{-}x}\sin\left(\pi x\right)[/latex] (Approximate answer to five decimal places.)

Find the volume generated by rotating the region bounded by the given curves about the specified line for the following exercises (31-34). Express the answers in exact form or approximate to the number of decimal places indicated.

31. [latex]y={e}^{\text{-}x}[/latex] [latex]y=0,x=-1x=0[/latex]; about [latex]x=1[/latex] (Express the answer in exact form.)
32. Find the area under the graph of [latex]y={\sec}^{3}x[/latex] from [latex]x=0\text{to}x=1[/latex]. (Round the answer to two significant digits.)

33. Find the area of the region enclosed by the curve [latex]y=x\cos{x}[/latex] and the x-axis for [latex]\frac{11\pi }{2}\le x\le \frac{13\pi }{2}[/latex]. (Express the answer in exact form.)

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

Trigonometric Integrals

For the following exercises (1-3), evaluate each of the following integrals by u-substitution.

1. [latex]\displaystyle\int {\sin}^{3}x\cos{x}dx[/latex]
2. [latex]\displaystyle\int {\tan}^{5}\left(2x\right){\sec}^{2}\left(2x\right)dx[/latex]
3. [latex]\displaystyle\int \tan\left(\dfrac{x}{2}\right){\sec}^{2}\left(\dfrac{x}{2}\right)dx[/latex]

In the following exercises (4-11), compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions.

4. [latex]\displaystyle\int {\sin}^{3}xdx[/latex]
5. [latex]\displaystyle\int \sin{x}\cos{x}dx[/latex]
6. [latex]\displaystyle\int {\sin}^{5}x{\cos}^{2}xdx[/latex]
7. [latex]\displaystyle\int \sqrt{\sin{x}}\cos{x}dx[/latex]
8. [latex]\displaystyle\int \sec{x}\tan{x}dx[/latex]
9. [latex]\displaystyle\int {\tan}^{2}x\sec{x}dx[/latex]
10. [latex]\displaystyle\int {\sec}^{4}xdx[/latex]
11. [latex]\displaystyle\int \csc{x}dx[/latex]

For the following exercise, find a general formula for the integrals.

12. [latex]\displaystyle\int {\sin}^{2}ax\cos{ax} dx[/latex]

Use the double-angle formulas to evaluate the following integrals (13-15).

13. [latex]{\displaystyle\int }_{0}^{\pi }{\sin}^{2}xdx[/latex]
14. [latex]\displaystyle\int {\cos}^{2}3xdx[/latex]
15. [latex]\displaystyle\int {\sin}^{2}xdx+\displaystyle\int {\cos}^{2}xdx[/latex]

For the following exercises (16-22), evaluate the definite integrals. Express answers in exact form whenever possible.

16. [latex]{\displaystyle\int }_{0}^{2\pi }\cos{x}\sin2xdx[/latex]
17. [latex]{\displaystyle\int }_{0}^{\pi }\cos\left(99x\right)\sin\left(101x\right)dx[/latex]
18. [latex]{\displaystyle\int }_{0}^{2\pi }\sin{x}\sin\left(2x\right)\sin\left(3x\right)dx[/latex]
19. [latex]{\displaystyle\int }_{\dfrac{\pi}{6}}^{\dfrac{\pi}{3}}\dfrac{{\cos}^{3}x}{\sqrt{\sin{x}}}dx[/latex] (Round this answer to three decimal places.)
20. [latex]{\displaystyle\int }_{0}^{\dfrac{\pi}{2}}\sqrt{1-\cos\left(2x\right)}dx[/latex]
21. Find the area of the region bounded by the graphs of the equations [latex]y={\cos}^{2}x,y={\sin}^{2}x,x=-\dfrac{\pi }{4},\text{and }x=\dfrac{\pi }{4}[/latex].
22. Find the average value of the function [latex]f\left(x\right)={\sin}^{2}x{\cos}^{3}x[/latex] over the interval [latex]\left[\text{-}\pi ,\pi \right][/latex].

For the following exercises (23-24), solve the differential equations.

23. [latex]\dfrac{dy}{d\theta }={\sin}^{4}\left(\pi \theta \right)[/latex]
24. Find the length of the curve [latex]y=\text{ln}\left(\sin{x}\right),\dfrac{\pi }{3}\le x\le \dfrac{\pi }{2}[/latex].

For the following exercises (25-26), use this information:

• The inner product of two functions [latex]f[/latex] and [latex]g[/latex] over [latex]\left[a,b\right][/latex] is defined by
  [latex]f\left(x\right)\cdot g\left(x\right)=\langle f,g\rangle ={\displaystyle\int }_{a}^{b}f\cdot gdx[/latex]
• Two distinct functions [latex]f[/latex] and [latex]g[/latex] are said to be orthogonal if [latex]\langle f,g\rangle =0[/latex].

25. Show that [latex]{\sin(2x),\cos(3x)}[/latex] are orthogonal over the interval [latex][\text{-}\pi ,\pi ][/latex].
26. Integrate [latex]{y}^{\prime }=\sqrt{\tan{x}}{\sec}^{4}x[/latex].

For the following pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning.

27. [latex]\displaystyle\int {\tan}^{350}x{\sec}^{2}xdx[/latex] or [latex]\displaystyle\int {\tan}^{350}x\sec{x}dx[/latex]

Simplify the following expressions by writing each one using a single trigonometric function.

28. [latex]9{\sec}^{2}\theta -9[/latex]
29. [latex]{a}^{2}+{a}^{2}{\text{sinh}}^{2}\theta[/latex]

Use the technique of completing the square to express each trinomial as the square of a binomial.

30. [latex]4{x}^{2}-4x+1[/latex]
31. [latex]\text{-}{x}^{2}-2x+4[/latex]

In the following exercises (32-38), integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

32. [latex]\displaystyle\int \frac{dx}{\sqrt{{x}^{2}-{a}^{2}}}[/latex]
33. [latex]\displaystyle\int \frac{dx}{\sqrt{1+9{x}^{2}}}[/latex]
34. [latex]\displaystyle\int \frac{dx}{{x}^{2}\sqrt{1-{x}^{2}}}[/latex]
35. [latex]\displaystyle\int \sqrt{{x}^{2}+9}dx[/latex]
36. [latex]\displaystyle\int \frac{{\theta }^{3}d\theta }{\sqrt{9-{\theta }^{2}}}d\theta[/latex]
37. [latex]\displaystyle\int \frac{dx}{{x}^{2}\sqrt{{x}^{2}+1}}[/latex]
38. [latex]{\displaystyle\int }_{-1}^{1}{\left(1-{x}^{2}\right)}^{\frac{3}{2}}dx[/latex]

In the following exercises (39-41), use the substitutions [latex]x=\text{sinh}\theta ,\text{cosh}\theta[/latex], or [latex]\text{tanh}\theta[/latex]. Express the final answers in terms of the variable [latex]x[/latex].

39. [latex]\displaystyle\int \frac{dx}{x\sqrt{1-{x}^{2}}}[/latex]
40. [latex]\displaystyle\int \frac{\sqrt{{x}^{2}-1}}{{x}^{2}}dx[/latex]
41. [latex]\displaystyle\int \frac{\sqrt{1+{x}^{2}}}{{x}^{2}}dx[/latex]

Use the technique of completing the square to evaluate the following integrals (42-43).

42. [latex]\displaystyle\int \frac{1}{{x}^{2}+2x+1}dx[/latex]
43. [latex]\displaystyle\int \frac{1}{\sqrt{\text{-}{x}^{2}+10x}}dx[/latex]

For the following exercises (44-49), solve each problem.

44. Evaluate the integral without using calculus: [latex]{\displaystyle\int }_{-3}^{3}\sqrt{9-{x}^{2}}dx[/latex].
45. Evaluate the integral [latex]\displaystyle\int \frac{dx}{\sqrt{1-{x}^{2}}}[/latex] using two different substitutions. First, let [latex]x=\cos\theta[/latex] and evaluate using trigonometric substitution. Second, let [latex]x=\sin\theta[/latex] and use trigonometric substitution. Are the answers the same?
46. Evaluate the integral [latex]\displaystyle\int \frac{x}{{x}^{2}+1}dx[/latex] using the form [latex]\displaystyle\int \frac{1}{u}du[/latex]. Next, evaluate the same integral using [latex]x=\tan\theta[/latex]. Are the results the same?
47. State the method of integration you would use to evaluate the integral [latex]\displaystyle\int {x}^{2}\sqrt{{x}^{2}-1}dx[/latex]. Why did you choose this method?
48. Find the length of the arc of the curve over the specified interval: [latex]y=\text{ln}x,\left[1,5\right][/latex]. Round the answer to three decimal places.
49. The region bounded by the graph of [latex]f\left(x\right)=\frac{1}{1+{x}^{2}}[/latex] and the [latex]x[/latex]-axis between [latex]x=0[/latex] and [latex]x=1[/latex] is revolved about the [latex]x[/latex]-axis. Find the volume of the solid that is generated.

For the following exercises (50-52), solve the initial-value problem for [latex]y[/latex] as a function of [latex]x[/latex].

50. [latex]\left(64-{x}^{2}\right)\frac{dy}{dx}=1,y\left(0\right)=3[/latex]
51. An oil storage tank can be described as the volume generated by revolving the area bounded by [latex]y=\frac{16}{\sqrt{64+{x}^{2}}},x=0,y=0,x=2[/latex] about the [latex]x[/latex]-axis. Find the volume of the tank (in cubic meters).
52. Find the length of the curve [latex]y=\sqrt{16-{x}^{2}}[/latex] between [latex]x=0[/latex] and [latex]x=2[/latex].

Partial Fractions

In the following exercises (1-7), express the rational function as a sum or difference of two simpler rational expressions.

1. [latex]\frac{{x}^{2}+1}{x\left(x+1\right)\left(x+2\right)}[/latex]
2. [latex]\frac{3x+1}{{x}^{2}}[/latex]
3. [latex]\frac{2{x}^{4}}{{x}^{2}-2x}[/latex]
4. [latex]\frac{1}{{x}^{2}\left(x - 1\right)}[/latex]
5. [latex]\frac{1}{x\left(x - 1\right)\left(x - 2\right)\left(x - 3\right)}[/latex]
6. [latex]\frac{3{x}^{2}}{{x}^{3}-1}=\frac{3{x}^{2}}{\left(x - 1\right)\left({x}^{2}+x+1\right)}[/latex]
7. [latex]\frac{3{x}^{4}+{x}^{3}+20{x}^{2}+3x+31}{\left(x+1\right){\left({x}^{2}+4\right)}^{2}}[/latex]

Use the method of partial fractions to evaluate each of the following integrals (8-12).

8. [latex]\displaystyle\int \frac{3x}{{x}^{2}+2x - 8}dx[/latex]
9. [latex]\displaystyle\int \frac{x}{{x}^{2}-4}dx[/latex]
10. [latex]\displaystyle\int \frac{2{x}^{2}+4x+22}{{x}^{2}+2x+10}dx[/latex]
11. [latex]\displaystyle\int \frac{2-x}{{x}^{2}+x}dx[/latex]
12. [latex]\displaystyle\int \frac{dx}{{x}^{3}-2{x}^{2}-4x+8}[/latex]

Evaluate the following integrals (13-14), which have irreducible quadratic factors.

13. [latex]\displaystyle\int \frac{2}{\left(x - 4\right)\left({x}^{2}+2x+6\right)}dx[/latex]
14. [latex]\displaystyle\int \frac{{x}^{3}+6{x}^{2}+3x+6}{{x}^{3}+2{x}^{2}}dx[/latex]

Use the method of partial fractions to evaluate the following integrals (15-16).

15. [latex]\displaystyle\int \frac{3x+4}{\left({x}^{2}+4\right)\left(3-x\right)}dx[/latex]
16. [latex]\displaystyle\int \frac{3x+4}{{x}^{3}-2x - 4}dx[/latex] (Hint: Use the rational root theorem.)

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.

17. [latex]\displaystyle\int \frac{{e}^{x}dx}{{e}^{2x}-{e}^{x}}dx[/latex]
18. [latex]\displaystyle\int \frac{\sin{x}}{{\cos}^{2}x+\cos{x} - 6}dx[/latex]
19. [latex]\displaystyle\int \frac{dt}{{\left({e}^{t}-{e}^{\text{-}t}\right)}^{2}}[/latex]
20. [latex]\displaystyle\int \frac{dx}{1+\sqrt{x+1}}[/latex]
21. [latex]\displaystyle\int \frac{\cos{x}}{\sin{x}\left(1-\sin{x}\right)}dx[/latex]
22. [latex]\underset{1}{\overset{2}{\displaystyle\int }}\frac{1}{{x}^{2}\sqrt{4-{x}^{2}}}dx[/latex]
23. [latex]\displaystyle\int \frac{1}{1+{e}^{x}}dx[/latex]

For the following exercises (44-49), use the given substitution to convert the integral to an integral of a rational function, then evaluate.

24. [latex]\displaystyle\int \frac{1}{\sqrt{x}+\sqrt[3]{x}}dx;x={u}^{6}[/latex]
25. Find the volume of the solid generated when the region bounded by [latex]y=\frac{1}{\sqrt{x\left(3-x\right)}}[/latex], [latex]y=0[/latex], [latex]x=1[/latex], and [latex]x=2[/latex] is revolved about the x-axis.

In the following exercises (1-7), solve the initial-value problem for [latex]x[/latex] as a function of [latex]t[/latex].

26. [latex]\left({t}^{2}-7t+12\right)\frac{dx}{dt}=1,\left(t>4,x\left(5\right)=0\right)[/latex]
27. [latex]\left(2{t}^{3}-2{t}^{2}+t - 1\right)\frac{dx}{dt}=3,x\left(2\right)=0[/latex]

For the following exercises (28-31), solve each problem.

28. Find the volume generated by revolving the area bounded by [latex]y=\frac{1}{{x}^{3}+7{x}^{2}+6x}x=1,x=7,\text{and }y=0[/latex] about the y-axis.
29. Evaluate the integral [latex]\displaystyle\int \frac{dx}{{x}^{3}+1}[/latex].
30. Find the area under the curve [latex]y=\frac{1}{1+\sin{x}}[/latex] between [latex]x=0[/latex] and [latex]x=\pi[/latex]. (Assume the dimensions are in inches.)
31. Evaluate [latex]\displaystyle\int \frac{\sqrt[3]{x - 8}}{x}dx[/latex].

Other Strategies for Integration

For the following exercises (1-8), use a table of integrals to evaluate the following integrals.

1. [latex]\displaystyle\int \frac{x+3}{{x}^{2}+2x+2}dx[/latex]
2. [latex]\displaystyle\int \frac{1}{\sqrt{{x}^{2}+6x}}dx[/latex]
3. [latex]\displaystyle\int x\cdot {2}^{{x}^{2}}dx[/latex]
4. [latex]\displaystyle\int \frac{dy}{\sqrt{4-{y}^{2}}}[/latex]
5. [latex]\displaystyle\int \csc\left(2w\right)\cot\left(2w\right)dw[/latex]
6. [latex]{\displaystyle\int }_{0}^{1}\frac{3xdx}{\sqrt{{x}^{2}+8}}[/latex]
7. [latex]{\displaystyle\int }_{0}^{\frac{\pi}{2}}{\tan}^{2}\left(\frac{x}{2}\right)dx[/latex]
8. [latex]\displaystyle\int {\tan}^{5}\left(3x\right)dx[/latex]

Use a CAS to evaluate the following integrals (9-14). Tables can also be used to verify the answers.

9. [latex]\displaystyle\int \frac{dw}{1+\sec\left(\frac{w}{2}\right)}[/latex]
10. [latex]{\displaystyle\int }_{0}^{t}\frac{dt}{4\cos{t}+3\sin{t}}[/latex]
11. [latex]\displaystyle\int \frac{dx}{{x}^{\frac{1}{2}}+{x}^{\frac{1}{3}}}[/latex]
12. [latex]\displaystyle\int {x}^{3}\sin{x}dx[/latex]
13. [latex]\displaystyle\int \frac{x}{1+{e}^{\text{-}{x}^{2}}}dx[/latex]
14. [latex]\displaystyle\int \frac{dx}{x\sqrt{x - 1}}[/latex]

Use a calculator or CAS to evaluate the following integrals (15-17).

15. [latex]{\displaystyle\int }_{0}^{\frac{z\pi}{4}}\cos\left(2x\right)dx[/latex]
16. [latex]{\displaystyle\int }_{0}^{8}\frac{2x}{\sqrt{{x}^{2}+36}}dx[/latex]
17. [latex]\displaystyle\int \frac{dx}{{x}^{2}+4x+13}[/latex]

Use tables to evaluate the following integrals (18-20). You may need to complete the square or change variables to put the integral into a form given in the table.

18. [latex]\displaystyle\int \frac{dx}{{x}^{2}+2x+10}[/latex]
19. [latex]\displaystyle\int \frac{{e}^{x}}{\sqrt{{e}^{2x}-4}}dx[/latex]
20. [latex]\displaystyle\int \frac{\text{arctan}\left({x}^{3}\right)}{{x}^{4}}dx[/latex]

For the following exercises (21-23), use tables to perform the integration.

21. [latex]\displaystyle\int \frac{dx}{\sqrt{{x}^{2}+16}}[/latex]
22. [latex]\displaystyle\int \frac{dx}{1-\cos\left(4x\right)}[/latex]

For the following exercises (23-27), solve each problem.

23. Find the area bounded by [latex]y\left(4+25{x}^{2}\right)=5,x=0,y=0,\text{and }x=4[/latex]. Use a table of integrals or a CAS.
24. Use substitution and a table of integrals to find the area of the surface generated by revolving the curve [latex]y={e}^{x},0\le x\le 3[/latex], about the [latex]x[/latex]-axis. (Round the answer to two decimal places.)
25. Use a CAS or tables to find the area of the surface generated by revolving the curve [latex]y=\cos{x},0\le x\le \frac{\pi }{2}[/latex], about the [latex]x[/latex]-axis. (Round the answer to two decimal places.)
26. Find the length of the curve [latex]y={e}^{x}[/latex] over [latex]\left[0,\text{ln}\left(2\right)\right][/latex].
27. Find the average value of the function [latex]f\left(x\right)=\frac{1}{{x}^{2}+1}[/latex] over the interval [latex]\left[-3,3\right][/latex].