1. True or False Integration by parts is based on the Product Rule for derivatives.

2. The integration by parts formula states that \(\int\! u\,dv\,{=}\) ______.

3. \(\int xe^{2x} dx\)

4. \(\int xe^{-3x} dx\)

5. \(\int x\cos x~dx\)

6. \(\int x\sin (3x)~dx\)

7. \(\int \sqrt{x}\ln x~dx\)

8. \(\int x^{-2}\ln x~dx\)

9. \(\int \cot^{-1}x~dx\)

10. \(\int \sin^{-1}x~dx\)

11. \(\int (\ln x)^{2} dx\)

12. \(\int x(\ln x)^{2} dx\)

13. \(\int x^{2} \sin x~dx\)

14. \(\int x^{2}cosx~dx\)

15. \(\int x\cos^{2}x~dx\)

16. \(\int x\sin^{2}x~dx\)

17. \(\int x\sinh x~dx\)

18. \(\int x\cosh x~dx\)

19. \(\int \cosh^{-1}x~dx\)

20. \(\int \sinh^{-1}x~dx\)

21. \(\int \sin(\ln x)~dx\)

22. \(\int \cos (\ln x)~dx\)

23. \(\int (\ln x)^{3} dx\)

24. \(\int (\ln x)^{4} dx\)

25. \(\int x^{2}(\ln x)^{2} dx\)

26. \(\int x^{3}(\ln x)^{2} dx\)

27. \(\int x^{2}\tan ^{-1}x~dx\)

28. \(\int x\tan ^{-1}x~dx\)

29. \(\int 7^{x} x~dx\)

30. \(\int 2^{-x} x~dx\)

31. \(\int_{0}^{\pi}e^{x}\cos x~dx\)

32. \(\int_{0}^{1}x^{2} e^{-x}dx\)

33. \(\int_{0}^{2}x^{2} e^{-3x} dx\)

34. \(\int_{0}^{\pi/4}x\tan ^{2}x~dx\)

35. \(\int_{1}^{9}\ln \sqrt{x} dx\)

36. \(\int_{\pi/4}^{3\pi/4}x\csc ^{2} x~dx\)

37. \(\int^{e}_1 (\ln x)^{2} dx\)

38. \(\int_{0}^{\pi/4}x \sec^{2} x~dx\)

39. \(f(x)=3\ln x\) and \(g(x)=x\ln x, x \ge 1\)

40. \(f(x)=4x\ln x\) and \(g(x)=x^{2}\ln x, x \ge 1\)

41. Area Under a Graph Find the area under the graph of \(y=e^{x}\sin x\) from \(0\) to \(\pi\).

42. Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of \(y=\cos x\) and the \(x\)-axis from \(x=0\) to \(x=\dfrac{\pi }{2}\) about the \(y\)-axis. See the figure below.

    

43. Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of \(y=\sin x\) and the \(x\)-axis from \(x=0\) to \(x=\dfrac{\pi }{2}\) about the \(y\)-axis.

    

44. Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of \(y=x \sqrt{\sin x}\) and the \(x\)-axis from \(x=0\) to \(x=\dfrac{\pi }{2}\) about the \(x\)-axis.

    

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45. Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of \(y=\ln x\) and the \(x\)-axis from \(x=1\) to \(x=e\) about the \(x\)-axis.

46. Area

    1. (a) Graph the functions \(f( x)=x^{3}e^{-3x}\) and \(g( x) =x^{2}e^{-3x}\) on the same set of coordinate axes.
    2. (b) Find the area enclosed by the graphs of \(f\) and \(g\).

47. Damped Spring The displacement \(x\) of a damped spring at time \(t\), \(0 \le t \le 5\), is given by \(x=x( t)=3e^{-t}\cos (2t) +2e^{-t}\sin (2t)\).

    1. (a) Graph \(x=x( t)\).
    2. (b) Find the least positive number \(t\) that satisfies \(x(t) =0\).
    3. (c) Find the area under the graph of \(x=x(t)\) from \(t=0\) to the value of \(t\) found in (b).

48. A function \(y=f( x)\) is continuous and differentiable on the interval \(( 2,6)\). If \(\int_{3}^{5}f( x)~dx=18\) and \(f( 3) =8\) and \(f( 5) =11,\) then find \(\int_{3}^{5}x f^{\prime}(x)~dx\).

49. \(\int \sin\sqrt{x}dx\)

50. \(\int e^{\sqrt{x}}dx\)

51. \(\int \cos x \ln(\sin x)~dx\)

52. \(\int e^{x}\ln (2+e^{x}) dx\)

53. \(\int e^{4x}\cos e^{2x} dx\)

54. \(\int \cos x\tan ^{-1} (\sin x)~dx\)

55. Find \(\int x^{3}e^{x^{2}} dx\). (Hint: Let \(u=x^{2},dv=xe^{x^{2}} dx\).)

56. Find \(\int x^{n} \ln x~dx\); \(n \neq -1\), \(n\) real.

57. Find \(\int xe^{x}\cos x~dx\).

58. Find \(\int xe^{x}\sin x~dx\).

59. \(\int x^{n}\sin ^{-1}x\,dx=\dfrac{x^{n+1}}{n+1}\sin^{-1}x-\dfrac{1}{n+1}\int \dfrac{x^{n+1}}{\sqrt{1-x^{2}}}\,dx\)

60. \(\int \dfrac{dx}{(x^{2}+1)^{n+1}}=\left( 1-\dfrac{1}{2n}\right) \int \dfrac{dx}{(x^{2}+1)^{n}}+\dfrac{x}{2n(x^{2}+1)^{n}}\)

61. \(\int \sin^{n}x~dx=-\dfrac{\sin ^{n-1}x\cos x}{n}+\dfrac{n-1}{n}\int \sin ^{n-2}x\,dx\)

62. \(\int \sin^{n}x \cos^{m} x~dx = -\dfrac{\sin^{n-1}x \cos^{m+1}x}{n+m} + \dfrac{n-1}{n+m}\int \sin^{n-2}x \cos^{m} xdx\)

    where \(m \neq -n\), \(m \neq -1\)

63. 1. (a) Find \(\int x^{2}e^{5x}\,dx\).
    2. (b) Using integration by parts, derive a reduction formula for \(\int x^{n}e^{kx}\,dx\), where \(k \neq 0\) and \(n \ge 2\) is an integer, in which the resulting integrand involves \(x^{n-1}\).

64. 1. (a) Assuming there is a function \(p\) for which \(\int x^{3}e^{x}dx\) \(=p( x) e^{x}\), show that \(p( x)+p^{\prime} (x) =x^{3}\).
    2. (b) Use integration by parts to find a polynomial \(p\) of degree 3 for which \(\int x^{3}e^{x}dx=p( x) e^{x}+C.\)

65. 1. (a) Use integration by parts with \(u=\sin x\) and \(dv=\cos x\,dx\) to find a function \(f\) for which \(\int \sin x\cos x~dx=f( x) +C_{1}\).
    2. (b) Use integration by parts with \(u=\cos x\) and \(dv=\sin x\,dx\) to find a function \(g\) for which \(\int \sin x\cos x~dx=g( x) +C_{2}\).
    3. (c) Use the trigonometric identity \(\sin (2x) =2\sin x\cos x\) and substitution to find a function \(h\) for which \[ \int \sin x\cos x~dx=h( x) +C_{3}.\]
    4. (d) Compare the functions \(f\) and \(g\). Find a relationship between \(C_{1}\) and \(C_{2}\).
    5. (e) Compare the functions \(f\) and \(h\). Find a relationship between \(C_{1}\) and \(C_{3}\).

66. Derive the formula \[ \int \ln (x+\sqrt{x^{2}+a^{2}})\,dx = x\ln (x+\sqrt{x^{2}+a^{2}}) -\sqrt{x^{2}+a^{2}}+C \]

67. Derive the formula \[ \int e^{ax}\sin (bx)~dx = \dfrac{e^{ax}[a\sin (bx)-b\cos (bx)]}{a^{2}+b^{2}} +C, a \gt 0, b\gt 0 \]

68. Suppose \(F(x)=\int_{0}^{x}t\,g^{\prime }(t)\,dt\) for all \(x \ge 0\). Show that \(F(x)=\) \(xg(x)-\int_{0}^{x}g(t)\,dt\).

69. Use Wallis’ formulas, given below, to find each definite integral.

    • \(\int_{0}^{\pi/2} \sin^{n}x~dx = \int_{0}^{\pi /2} \cos^{n}x~dx\) \(n \gt 1\) an integer \[=\left\{ \begin{array}{@{}l@{\qquad}l} \dfrac{(n-1)(n-3)\cdots (4)(2)}{n(n-2)\cdots (5)(3)(1)} & n \gt 1 \hbox{ is odd} \\ \dfrac{(n-1)(n-3)\cdots (5)(3)(1)}{n(n-2)\cdots (4)(2)}\left( \dfrac{\pi }{2} \right) & n\gt1\hbox{ is even} \end{array} \right.\]

    1. (a) \(\int_{0}^{\pi /2}\sin ^{6}x~dx\)
    2. (b) \(\int_{0}^{\pi /2}\sin ^{5}x~dx\)
    3. (c) \(\int_{0}^{\pi /2}\cos ^{8}x~dx\)
    4. (d) \(\int_{0}^{\pi /2}\cos ^{6}x~dx\)

Challenge Problems

70. Derive Wallis’ formulas given in Problem 69. (Hint: Use the result of Problem 61.)

71. 1. (a) If \(n\) is a positive integer, use integration by parts to show that there is a polynomial \(p\) of degree \(n\) for which \[ \int x^{n}e^{x}dx=p( x) e^{x}+C \]

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    2. (b) Show that \(p( x) +p^{\prime} ( x) =x^{n}\).
    3. (c) Show that \(p\) can be written in the form \[ p(x) = \sum \limits_{k=0}^{n}(-1)^{k} \dfrac{n!}{(n-k)!}x^{n-k} \]

72. Show that for any positive integer \(n\), \[ \begin{eqnarray*} \hspace{-20pc}\int_{0}^{1}e^{x^{2}} dx \\ = e\cdot \!\left[ 1-\frac{2}{3}+\frac{4}{15}-\frac{8}{105}+\cdots +\frac{(-1)^{n}2^{n}}{(2n+1)(2n-1)\cdots 3\cdot 1}\right] \\+ (-1)^{n+1} \cdot \frac{2^{n+1}}{(2n+1)(2n-1)\cdots 3\cdot 1} \int_{0}^{1}x^{2n+2}e^{x^{2}}dx \end{eqnarray*} \]

73. Use integration by parts to show that if \(f\) is a polynomial of degree \(n \ge 1\), then \(\int f( x) e^{x}dx=g( x) e^{x}+C\) for some polynomial \(g( x)\) of degree \(n\).

74. Start with the identity \(f(b)-f(a)=\int_{a}^{b}f^{\prime}(t)\,dt\) and derive the following generalizations of the Mean Value Theorem for Integrals:

    • (a) \(f(b)-f(a)=f^{\prime} (a)(b-a)- \int_{a}^{b}f^{\prime \prime}(t)(t-b)\,dt\)
    • (b) \( f(b)-f(a)=f^{\prime} (a)(b-a)+\dfrac{f^{\prime\prime} (a)}{2}(b-a)^{2} + \int_{a}^{b}\dfrac{f^{\prime \prime \prime} (t)}{2}(t-b)^{2} dt \)

75. If \(y=f(x)\) has the inverse function given by \(x=f^{-1}(y)\), show that \[ \begin{equation*} \int_{a}^{b}f(x)\,dx+\int_{f(a)}^{f(b)}f^{-1}(y)\,dy=bf(b)-af(a) \end{equation*} \]

76. 1. (a) When integration by parts is used to find \(\int e^{x}\cosh x~dx\), what happens? Explain.
    2. (b) Find \(\int e^{x}\cosh x~dx\) without using integration by parts.