# Chapter 8

## Exercises 8.1

1. 1.

T

2. 3.

$\sin x-x\cos x+C$

3. 5.

$-x^{2}\cos x+2x\sin x+2\cos x+C$

4. 7.

$1/2e^{x^{2}}+C$

5. 9.

$-\frac{1}{2}xe^{-2x}-\frac{e^{-2x}}{4}+C$

6. 11.

$1/5e^{2x}(\sin x+2\cos x)+C$

7. 13.

$\frac{1}{10}e^{5x}(\sin(5x)+\cos(5x))+C$

8. 15.

$\sqrt{1-x^{2}}+x\sin^{-1}(x)+C$

9. 17.

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

10. 19.

$\frac{1}{2}x^{2}\ln\left\lvert x\right\rvert-\frac{x^{2}}{4}+C$

11. 21.

$-\frac{x^{2}}{4}+\frac{1}{2}x^{2}\ln\left\lvert x-1\right\rvert-\frac{x}{2}-% \frac{1}{2}\ln\left\lvert x-1\right\rvert+C$

12. 23.

$\frac{1}{3}x^{3}\ln\left\lvert x\right\rvert-\frac{x^{3}}{9}+C$

13. 25.

$2x+x\left(\ln\left\lvert x+1\right\rvert\right)^{2}+\left(\ln\left\lvert x+1% \right\rvert\right)^{2}-2x\ln\left\lvert x+1\right\rvert-2\ln\left\lvert x+1% \right\rvert+2+C$

14. 27.

$\ln\left\lvert\sin x\right\rvert-x\cot x+C$

15. 29.

$\frac{1}{3}(x^{2}-2)^{3/2}+C$

16. 31.

$x\sec x-\ln\left\lvert\sec x+\tan x\right\rvert+C$

17. 33.

$x\sinh x-\cosh x+C$

18. 35.

$x\sinh^{-1}x-\sqrt{x^{2}+1}+C$

19. 37.

$1/2x\big{(}\sin(\ln x)-\cos(\ln x)\big{)}+C$

20. 39.

$\frac{1}{2}x\ln|x|-\frac{x}{2}+C$

21. 41.

$1/2x^{2}+C$

22. 43.

$\pi$

23. 45.

$0$

24. 47.

$1/2$

25. 49.

$\frac{3}{4e^{2}}-\frac{5}{4e^{4}}$

26. 51.

$\frac{1}{5}\left(e^{\pi}+e^{-\pi}\right)$

27. 53.
28. 55.
(a) $b_{n}=(-1)^{n+1}2/n$ (b) answers will vary

## Exercises 8.2

1. 1.

F

2. 3.

F

3. 5.

$\frac{1}{4}\sin^{4}x+C$

4. 7.

$\frac{3}{8}x+\frac{1}{4}\sin 2x+\frac{1}{32}\sin 4x+C$

5. 9.

$\frac{1}{6}\cos^{6}x-\frac{1}{4}\cos^{4}x+C$

6. 11.

$\frac{1}{2}\cos^{2}x-\ln\left\lvert\cos x\right\rvert+C$

7. 13.

$\left(\frac{2}{7}\cos^{3}x-\frac{2}{3}\cos x\right)\sqrt{\cos x}+C$

8. 15.

$\frac{1}{2}\left(\frac{1}{4}\sin(4x)-\frac{1}{10}\sin(10x)\right)+C$

9. 17.

$\frac{1}{2}\left(\sin(x)+\frac{1}{3}\sin(3x)\right)+C$

10. 19.

$\tan x-x+C$

11. 21.

$\frac{\tan^{6}(x)}{6}+\frac{\tan^{4}x}{4}+C$

12. 23.

$\frac{\sec^{5}(x)}{5}-\frac{\sec^{3}x}{3}+C$

13. 25.

$\frac{1}{3}\tan^{3}x-\tan x+x+C$

14. 27.

$\frac{1}{2}\left(\sec x\tan x-\ln\left\lvert\sec x+\tan x\right\rvert\right)+C$

15. 29.

$\ln\left\lvert\csc x-\cot x\right\rvert+C$

16. 31.

$-\frac{1}{2}\cot^{2}x+\ln\left\lvert\csc x\right\rvert+C$

17. 33.

$\frac{2}{5}$

18. 35.

$32/315$

19. 37.

$2/3$

20. 39.

$16/15$

21. 41.

1

## Exercises 8.3

1. 1.

backwards

2. 3.
(a) $\tan^{2}\theta+1=\sec^{2}\theta$ (b) $9\sec^{2}\theta$.
3. 5.

$\frac{1}{2}\left(x\sqrt{x^{2}+1}+\ln\left\lvert\sqrt{x^{2}+1}+x\right\rvert% \right)+C$

4. 7.

$x\sqrt{x^{2}+1/4}+\frac{1}{4}\ln\left\lvert 2\sqrt{x^{2}+1/4}+2x\right\rvert+C% =\frac{1}{2}x\sqrt{4x^{2}+1}+\frac{1}{4}\ln\left\lvert\sqrt{4x^{2}+1}+2x\right% \rvert+C$

5. 9.

$4\left(\frac{1}{2}x\sqrt{x^{2}-1/16}-\frac{1}{32}\ln\left\lvert 4x+4\sqrt{x^{2% }-1/16}\right\rvert\right)+C=\frac{1}{2}x\sqrt{16x^{2}-1}-\frac{1}{8}\ln\left% \lvert 4x+\sqrt{16x^{2}-1}\right\rvert+C$

6. 11.

$3\sin^{-1}\left(\frac{x}{\sqrt{7}}\right)+C$ (Trig. Subst. is not needed)

7. 13.

$2\left(\frac{x}{4}\sqrt{x^{2}+4}+\ln\left\lvert\frac{\sqrt{x^{2}+1}}{2}+\frac{% x}{2}\right\rvert\right)+C$

8. 15.

$\frac{1}{2}\left(9\sin^{-1}(x/3)+x\sqrt{9-x^{2}}\right)+C$

9. 17.

$\sqrt{7}\tan^{-1}\left(\frac{x}{\sqrt{7}}\right)+C$

10. 19.

$14\sin^{-1}\left(\frac{x}{\sqrt{5}}\right)+C$

11. 21.

$\frac{5}{4}\sec^{-1}(\left\lvert x\right\rvert/4)+C$

12. 23.

$\frac{\tan^{-1}\left(\frac{x-1}{\sqrt{7}}\right)}{\sqrt{7}}+C$

13. 25.

$3\sin^{-1}\left(\frac{x-4}{5}\right)+C$

14. 27.

$\sqrt{x^{2}-11}-\sqrt{11}\sec^{-1}(x/\sqrt{11})+C$

15. 29.

$-\frac{1}{\sqrt{x^{2}+9}}+C$ (Trig. Subst. is not needed)

16. 31.

$\frac{1}{18}\frac{x+2}{x^{2}+4x+13}+\frac{1}{54}\tan^{-1}\left(\frac{x+2}{3}% \right)+C$

17. 33.

$\frac{1}{7}\left(-\frac{\sqrt{5-x^{2}}}{x}-\sin^{-1}(x/\sqrt{5})\right)+C$

18. 35.

$\pi/2$

19. 37.

$2\sqrt{2}+2\ln(1+\sqrt{2})$

20. 39.

$9\sin^{-1}(1/3)+2\sqrt{2}$ Note: the new bounds of integration are $\sin^{-1}(-1/3)<\theta<\sin^{-1}(1/3)$. The final answer comes with recognizing that $\sin^{-1}(-1/3)=-\sin^{-1}(1/3)$ and that $\cos\big{(}\sin^{-1}(1/3)\big{)}=\cos\big{(}\sin^{-1}(-1/3)\big{)}=2\sqrt{2}/3$.

21. 41.
(a) $\pi(1-\frac{\pi}{4})$ (b) $\pi(\sqrt{2}-\ln(1+\sqrt{2}))$

## Exercises 8.4

1. 1.

rational

2. 3.

$\frac{A}{x}+\frac{B}{x-3}$

3. 5.

$\frac{A}{x-\sqrt{7}}+\frac{B}{x+\sqrt{7}}$

4. 7.

$3\ln\left\lvert x-2\right\rvert+4\ln\left\lvert x+5\right\rvert+C$

5. 9.

$\frac{1}{3}(\ln\left\lvert x+2\right\rvert-\ln\left\lvert x-2\right\rvert)+C$

6. 11.

$-\frac{4}{x+8}-3\ln\left\lvert x+8\right\rvert+C$

7. 13.

$-\ln\left\lvert 2x-3\right\rvert+5\ln\left\lvert x-1\right\rvert+2\ln\left% \lvert x+3\right\rvert+C$

8. 15.

$x+\ln\left\lvert x-1\right\rvert-\ln\left\lvert x+2\right\rvert+C$

9. 17.

$2x+C$

10. 19.

$\frac{1}{x}+\frac{1}{2}\ln\left\lvert\frac{x-1}{x+1}\right\rvert+C$

11. 21.

$\ln\left\lvert 3x^{2}+5x-1\right\rvert+2\ln\left\lvert x+1\right\rvert+C$

12. 23.

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

13. 25.

$\frac{1}{2}\left(3\ln\left\lvert x^{2}+2x+17\right\rvert-4\ln\left\lvert x-7% \right\rvert+\tan^{-1}\left(\frac{x+1}{4}\right)\right)+C$

14. 27.

$-\frac{1}{4}\ln(x^{2}+3)+\frac{1}{4}\ln(x^{2}+1)+C=\frac{1}{4}\ln\frac{x^{2}+1% }{x^{2}+3}+C$

15. 29.

$3\left(\ln\left\lvert x^{2}-2x+11\right\rvert+\ln\left\lvert x-9\right\rvert% \right)+3\sqrt{\frac{2}{5}}\tan^{-1}\left(\frac{x-1}{\sqrt{10}}\right)+C$

16. 31.

$\frac{1}{32}\ln\left\lvert x-2\right\rvert-\frac{1}{32}\ln\left\lvert x+2% \right\rvert-\frac{1}{16}\tan^{-1}(x/2)+C$

17. 33.

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

18. 35.

$\ln(2000/243)\approx 2.108$

19. 37.

$-\pi/4+\tan^{-1}3-\ln(11/9)\approx 0.263$

20. 39.

## Exercises 8.5

1. 1.

$x\sin^{-1}x+\sqrt{1-x^{2}}+C$

2. 3.

$\displaystyle 18\ln\left\lvert x-2\right\rvert-9\ln\left\lvert x-1\right\rvert% -5\ln\left\lvert x-3\right\rvert+C$

3. 5.

$\displaystyle\frac{x}{25\sqrt{x^{2}+25}}+C$

4. 7.

$\displaystyle 2\ln\left\lvert x-1\right\rvert-\ln\left\lvert x\right\rvert-% \frac{1}{x-1}-\frac{1}{(x-1)^{2}}+C$

5. 9.

$\displaystyle\frac{1}{2}e^{x^{2}}(x^{2}-1)+C$

6. 11.

$\displaystyle\frac{1}{13}e^{2x}(2\sin 3x-3\cos 3x)+C$

7. 13.

$\displaystyle-\sqrt{4-x^{2}}+C$

8. 15.

$\displaystyle 2\tan^{-1}\sqrt{x}+C$

9. 17.

$\displaystyle\frac{1}{27}[6x\sin 3x-(9x^{2}-2)\cos 3x]+C$

10. 19.

$\displaystyle\frac{2}{3}(1+e^{x})^{3/2}+C$

11. 21.

$\displaystyle\frac{1}{3}\tan^{3}x+C$

12. 23.

$\displaystyle-\frac{1}{4}(8-x^{3})^{4/3}+C$

13. 25.

$\displaystyle\frac{1}{10}(3-2x)^{5/2}-\frac{1}{2}(3-2x)^{3/2}+C$

14. 27.

$\displaystyle\frac{2}{5}x^{5/2}-\frac{8}{3}x^{3/2}+6x^{1/2}+C$

15. 29.

$\displaystyle\frac{11}{2}\ln\left\lvert x+5\right\rvert-\frac{15}{2}\ln\left% \lvert x+7\right\rvert+C$

16. 31.

$\displaystyle e^{\tan x}+C$

17. 33.

$\displaystyle-\frac{1}{5}\cot^{5}x+\frac{1}{3}\cot^{3}x-\cot x-x+C$

18. 35.

$\displaystyle\frac{1}{3}x^{3}-\frac{1}{4}\tanh 4x+C$

19. 37.

$\displaystyle 3\sin^{-1}\bigl{(}\frac{x+5}{6}\bigr{)}+C$

20. 39.

$\displaystyle\frac{1}{3}\sec^{3}x-\sec x+C$

21. 41.

$\displaystyle-2\sin^{-1}\bigl{(}\frac{2x}{3}\bigr{)}-\frac{1}{x}\sqrt{9-4x^{2}% }+C$

22. 43.

$\displaystyle-\ln x+\frac{4}{\sqrt{x}}+4\ln\left\lvert 1-\sqrt{x}\right% \rvert+C$

23. 45.

$\displaystyle\frac{-x}{2(25+x^{2})}+\frac{1}{10}\tan^{-1}\bigl{(}\frac{x}{5}% \bigr{)}+C$

24. 47.

$\displaystyle\frac{1}{4}x^{4}-2x^{2}+4\ln\left\lvert x\right\rvert+C$

25. 49.

$\displaystyle\frac{3}{64}(2x+3)^{8/3}-\frac{9}{20}(2x+3)^{5/3}+\frac{27}{16}(2% x+3)^{2/3}+C$

26. 51.

$\displaystyle-\frac{1}{7}\cos 7x+C$

27. 53.
28. 55.

$\frac{1}{2}\ln\left\lvert\tan\frac{\theta}{2}\right\rvert-\frac{1}{4}\tan^{2}% \frac{\theta}{2}+C$.

## Exercises 8.6

1. 1.

The interval of integration is finite, and the integrand is continuous on that interval.

2. 3.

converges; could also state $\leq 10$.

3. 5.

$p>1$

4. 7.

$e^{5}/2$

5. 9.

$1/3$

6. 11.

$1/\ln 2$

7. 13.

diverges

8. 15.

$1$

9. 17.

diverges

10. 19.

diverges

11. 21.

$2\sqrt{3}$

12. 23.

diverges

13. 25.

diverges

14. 27.

$1$

15. 29.

$0$

16. 31.

$-1/4$

17. 33.

$-1$

18. 35.

diverges

19. 37.

diverges; Limit Comparison Test with $1/x$.

20. 39.

diverges; Limit Comparison Test with $1/x$.

21. 41.

converges; Direct Comparison Test with $e^{-x}$.

22. 43.

converges; Direct Comparison Test with $1/(x^{2}-1)$.

23. 45.

converges; Direct Comparison Test with $1/e^{x}$.

24. 47.
(a) $e^{-\lambda a}$ (b) $\frac{1}{\lambda}$ (c) $e^{-1}$

## Exercises 8.7

1. 1.

F

2. 3.

They are superseded by the Trapezoidal Rule; it takes an equal amount of work and is generally more accurate.

3. 5.
(a) $3/4$ (b) $2/3$ (c) $2/3$
4. 7.
(a) $\frac{1}{4}(1+\sqrt{2})\pi\approx 1.896$ (b) $\frac{1}{6}(1+2\sqrt{2})\pi\approx 2.005$ (c) $2$
5. 9.
(a) $38.5781$ (b) $147/4\approx 36.75$ (c) $147/4\approx 36.75$
6. 11.
(a) $0$ (b) $0$ (c) $0$
7. 13.
Trapezoidal Rule: $0.9006$ Simpson’s Rule: $0.90452$
8. 15.
Trapezoidal Rule: $13.9604$ Simpson’s Rule: $13.9066$
9. 17.
Trapezoidal Rule: $1.1703$ Simpson’s Rule: $1.1873$
10. 19.
Trapezoidal Rule: $1.0803$ Simpson’s Rule: $1.077$
11. 21.
(a) $n=161$ (using $\max\big{(}f\,^{\prime\prime}(x)\big{)}=1$) (b) $n=12$ (using $\max\big{(}f\,^{(4)}(x)\big{)}=1$)
12. 23.
(a) $n=1004$ (using $\max\big{(}f\,^{\prime\prime}(x)\big{)}=39$) (b) $n=62$ (using $\max\big{(}f\,^{(4)}(x)\big{)}=800$)
13. 25.
(a) Area is $30.8667$ cm${}^{2}$. (b) Area is $308,667$ yd${}^{2}$.
14. 27.

Let $f(x)=a(x-x_{1})^{2}+b(x-x_{1})+c$, so that $f(x_{1})=c=y_{1}$, $f(x_{1}+\Delta x)=a\Delta x^{2}+b\Delta x+c=y_{2}$, and $f(x_{1}+2\Delta x)=4a\Delta x^{2}+2b\Delta x+c=y_{3}$. Therefore, $a=\frac{y_{1}-2y_{2}+y_{3}}{2(\Delta x)^{2}}$ and $b=\frac{4y_{2}-y_{3}-3y_{1}}{2\Delta x}$, and $\displaystyle\int_{x_{1}}^{x_{1}+2\Delta x}a(x-x_{1})^{2}+b(x-x_{1})+c% \operatorname{d}\!x=\frac{a(2\Delta x)^{3}}{3}+\frac{b(2\Delta x)^{2}}{2}+c(2% \Delta x)=\frac{4(y_{1}-2y_{2}+y_{3})\Delta x}{3}+(4y_{2}-y_{3}-3y_{1})\Delta x% +2y_{1}\Delta x=\frac{\Delta x}{3}(4y_{1}-8y_{2}+4y_{3}+12y_{2}-3y_{3}-9y_{1}+% 6y_{1})=\frac{\Delta x}{3}(y_{1}+4y_{2}+y_{3})$. 