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.