Solutions To Selected Problems Chapter G Chapter I

Chapter H

Exercises H.1

1.

T
2.

F
3.

$\sin x-x\cos x+C$
4.

$-e^{-x}-xe^{-x}+C$
5.

$-x^{2}\cos x+2x\sin x+2\cos x+C$
6.

$-x^{3}\cos x+3x^{2}\sin x+6x\cos x-6\sin x+C$
7.

$1/2e^{x^{2}}+C$
8.

$x^{3}e^{x}-3x^{2}e^{x}+6xe^{x}-6e^{x}+C$
9.

$-\frac{1}{2}xe^{-2x}-\frac{e^{-2x}}{4}+C$
10.

$\frac{1}{2}e^{x}(\sin x-\cos x)+C$
11.

$1/5e^{2x}(\sin x+2\cos x)+C$
12.

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

$\frac{1}{10}e^{5x}(\sin(5x)+\cos(5x))+C$
14.

$-\frac{1}{2}\cos^{2}x+C$
15.

$\sqrt{1-x^{2}}+x\sin^{-1}(x)+C$
16.

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

$\frac{1}{2}x^{2}\tan^{-1}(x)-\frac{x}{2}+\frac{1}{2}\tan^{-1}(x)+C$
18.

$x\cos^{-1}x-\sqrt{1-x^{2}}+C$
19.

$\frac{1}{2}x^{2}\ln\left\lvert x\right\rvert-\frac{x^{2}}{4}+C$
20.

$-\frac{x^{2}}{4}+\frac{1}{2}x^{2}\ln\left\lvert x\right\rvert-2x\ln\left\lvert x% \right\rvert+2x+C$
21.

$-\frac{x^{2}}{4}+\frac{1}{2}x^{2}\ln\left\lvert x-1\right\rvert-\frac{x}{2}-% \frac{1}{2}\ln\left\lvert x-1\right\rvert+C$
22.

$\frac{1}{2}x^{2}\ln\left(x^{2}\right)-\frac{x^{2}}{2}+C$
23.

$\frac{1}{3}x^{3}\ln\left\lvert x\right\rvert-\frac{x^{3}}{9}+C$
24.

$2x+x\left(\ln\left\lvert x\right\rvert\right)^{2}-2x\ln\left\lvert x\right% \rvert+C$
25.

$2x+x\left(\ln\left\lvert x+1\right\rvert\right)^{2}+\left(\ln\left\lvert x+1% \right\rvert\right)^{2}-2x\ln\left\lvert x+1\right\rvert-2\ln\left\lvert x+1% \right\rvert+2+C$
26.

$x\tan x+\ln\left\lvert\cos x\right\rvert+C$
27.

$\ln\left\lvert\sin x\right\rvert-x\cot x+C$
28.

$\frac{2}{5}(x-2)^{5/2}+\frac{4}{3}(x-2)^{3/2}+C$
29.

$\frac{1}{3}(x^{2}-2)^{3/2}+C$
30.

$\sec x+C$
31.

$x\sec x-\ln\left\lvert\sec x+\tan x\right\rvert+C$
32.

$-x\csc x-\ln\left\lvert\csc x+\cot x\right\rvert+C$
33.

$x\sinh x-\cosh x+C$
34.

$x\cosh x-\sinh x+C$
35.

$x\sinh^{-1}x-\sqrt{x^{2}+1}+C$
36.

$x\tanh^{-1}x+\frac{1}{2}\ln\left\lvert x^{2}-1\right\rvert+C$
37.

$1/2x\big(\sin(\ln x)-\cos(\ln x)\big)+C$
38.

$2\sin\left(\sqrt{x}\right)-2\sqrt{x}\cos\left(\sqrt{x}\right)+C$
39.

$\frac{1}{2}x\ln|x|-\frac{x}{2}+C$
40.

$2\sqrt{x}e^{\sqrt{x}}-2e^{\sqrt{x}}+C$
41.

$1/2x^{2}+C$
42.

$\frac{1}{2}e^{x^{2}}(x^{2}-1)+C$
43.

$\pi$
44.

$-2/e$
45.

$0$
46.

$\frac{3\pi^{2}}{2}-12$
47.

$1/2$
48.

$6-2e$
49.

$\frac{3}{4e^{2}}-\frac{5}{4e^{4}}$
50.

$\frac{1}{2}+\frac{e^{\pi}}{2}$
51.

$\frac{1}{5}\left(e^{\pi}+e^{-\pi}\right)$
52.

$\frac{3}{13}(1+e^{2\pi/3})+C$
53.
54.
- $\pi(e-2)$
  
  $\frac{\pi}{2}(e^{2}+1)$
55.
- $b_{n}=(-1)^{n+1}2/n$
  
  answers will vary
56.
- $b_{n}=(-1)^{(n-1)/2}4/\pi n^{2}$ for odd $n$ and $b_{n}=0$ for even $n$
  
  answers will vary

Exercises H.2

1.

F
2.

F
3.

F
4.

F
5.

$\frac{1}{4}\sin^{4}x+C$
6.

$\frac{1}{2}x+\frac{1}{4}\sin 2x+C$
7.

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

$\frac{1}{5}\cos^{5}x-\frac{1}{3}\cos^{3}x+C$
9.

$\frac{1}{6}\cos^{6}x-\frac{1}{4}\cos^{4}x+C$
10.

$\frac{1}{11}\sin^{11}x-\frac{2}{9}\sin^{9}x+\frac{1}{7}\sin^{7}x+C$
11.

$\frac{1}{2}\cos^{2}x-\ln\left\lvert\cos x\right\rvert+C$
12.

$\frac{x}{8}-\frac{1}{32}\sin(4x)+C$
13.

$\left(\frac{2}{7}\cos^{3}x-\frac{2}{3}\cos x\right)\sqrt{\cos x}+C$
14.

$\frac{1}{2}\left(-\frac{1}{3}\cos(3x)+\cos(-x)\right)+C$
15.

$\frac{1}{2}\left(\frac{1}{4}\sin(4x)-\frac{1}{10}\sin(10x)\right)+C$
16.

$\frac{1}{2}\left(\frac{1}{\pi}\sin(\pi x)-\frac{1}{3\pi}\sin(3\pi x)\right)+C$
17.

$\frac{1}{2}\left(\sin(x)+\frac{1}{3}\sin(3x)\right)+C$
18.

$\frac{1}{\pi}\sin(\frac{\pi}{2}x)+\frac{1}{3\pi}\sin(\pi x)+C$
19.

$\tan x-x+C$
20.

$\frac{1}{5}\tan^{5}x+\frac{1}{3}\tan^{3}x+C$
21.

$\frac{\tan^{6}(x)}{6}+\frac{\tan^{4}x}{4}+C$
22.

$\frac{\tan^{4}(x)}{4}+C$
23.

$\frac{\sec^{5}(x)}{5}-\frac{\sec^{3}x}{3}+C$
24.

$\frac{\sec^{9}(x)}{9}-\frac{2\sec^{7}x}{7}+\frac{\sec^{5}x}{5}+C$
25.

$\frac{1}{3}\tan^{3}x-\tan x+x+C$
26.

$\frac{1}{4}\tan x\sec^{3}x+\frac{3}{8}\left(\sec x\tan x+\ln\left\lvert\sec x+% \tan x\right\rvert\right)+C$
27.

$\frac{1}{2}\left(\sec x\tan x-\ln\left\lvert\sec x+\tan x\right\rvert\right)+C$
28.

$\frac{1}{4}\tan x\sec^{3}x-\frac{1}{8}\left(\sec x\tan x+\ln\left\lvert\sec x+% \tan x\right\rvert\right)+C$
29.

$\ln\left\lvert\csc x-\cot x\right\rvert+C$
30.

$\frac{1}{3}\csc^{3}x-\frac{1}{5}\csc^{5}x+C$
31.

$-\frac{1}{2}\cot^{2}x+\ln\left\lvert\csc x\right\rvert+C$
32.

$-\frac{1}{9}\cot^{9}x-\frac{1}{7}\cot^{7}x+C$
33.

$\frac{2}{5}$
34.

$0$
35.

$32/315$
36.

$1/2$
37.

$2/3$
38.

$1/5$
39.

$16/15$
40.

$\sqrt{3}-\frac{\pi}{3}$
41.

1

Exercises H.3

1.

backwards
2.

$5\sin\theta$
3.
- $\tan^{2}\theta+1=\sec^{2}\theta$
  
  $9\sec^{2}\theta$ .
4.

Because we are considering $a>0$ and $x=a\sin\theta$ , which means $\theta=\sin^{-1}(x/a)$ . The arcsine function has a domain of $-\pi/2\leq\theta\leq\pi/2$ ; on this domain, $\cos\theta\geq 0$ , so $a\cos\theta$ is always non-negative, allowing us to drop the absolute value signs.
5.

$\frac{1}{2}\left(x\sqrt{x^{2}+1}+\ln\left\lvert\sqrt{x^{2}+1}+x\right\rvert% \right)+C$
6.

$\frac{1}{2}x\sqrt{x^{2}-1}-\frac{1}{2}\ln\left\lvert x+\sqrt{x^{2}-1}\right% \rvert+C$
7.

$x\sqrt{x^{2}+1/4}+\frac{1}{4}\ln\left\lvert 2\sqrt{x^{2}+1/4}+2x\right\rvert+C% =\frac{1}{2}x\sqrt{4x^{2}+1}+\frac{1}{4}\ln\left\lvert\sqrt{4x^{2}+1}+2x\right% \rvert+C$
8.

$\frac{1}{6}\sin^{-1}(3x)+\frac{3}{2}\sqrt{1/9-x^{2}}+C=\frac{1}{6}\sin^{-1}(3x% )+\frac{1}{2}\sqrt{1-9x^{2}}+C$
9.

$4\left(\frac{1}{2}x\sqrt{x^{2}-1/16}-\frac{1}{32}\ln\left\lvert 4x+4\sqrt{x^{2% }-1/16}\right\rvert\right)+C=\frac{1}{2}x\sqrt{16x^{2}-1}-\frac{1}{8}\ln\left% \lvert 4x+\sqrt{16x^{2}-1}\right\rvert+C$
10.

$8\ln\left\lvert\frac{\sqrt{x^{2}+2}}{\sqrt{2}}+\frac{x}{\sqrt{2}}\right\rvert+C$ ; with Section 7.4, we can state the answer as $8\sinh^{-1}(x/\sqrt{2})+C$ .
11.

$3\sin^{-1}\left(\frac{x}{\sqrt{7}}\right)+C$ (Trig. Subst. is not needed)
12.

$5\ln\left\lvert\frac{x}{\sqrt{8}}+\frac{\sqrt{x^{2}-8}}{\sqrt{8}}\right\rvert+C$
13.

$2\left(\frac{x}{4}\sqrt{x^{2}+4}+\ln\left\lvert\frac{\sqrt{x^{2}+1}}{2}+\frac{% x}{2}\right\rvert\right)+C$
14.

$\frac{1}{2}\left(\sin^{-1}x+x\sqrt{1-x^{2}}\right)+C$
15.

$\frac{1}{2}\left(9\sin^{-1}(x/3)+x\sqrt{9-x^{2}}\right)+C$
16.

$\frac{1}{2}x\sqrt{x^{2}-16}-8\ln\left\lvert\frac{x}{4}+\frac{\sqrt{x^{2}-16}}{% 4}\right\rvert+C$
17.

$\sqrt{7}\tan^{-1}\left(\frac{x}{\sqrt{7}}\right)+C$
18.

$3\sin^{-1}\left(\frac{x}{3}\right)+C$
19.

$14\sin^{-1}\left(\frac{x}{\sqrt{5}}\right)+C$
20.

$\frac{2}{3}\sec^{-1}(\left\lvert x\right\rvert/3)+C$
21.

$\frac{5}{4}\sec^{-1}(\left\lvert x\right\rvert/4)+C$
22.

$\frac{1}{2}\sin^{-1}\left(x^{2}\right)+C$
23.

$\frac{\tan^{-1}\left(\frac{x-1}{\sqrt{7}}\right)}{\sqrt{7}}+C$
24.

$2\sin^{-1}\left(\frac{x-3}{4}\right)+C$
25.

$3\sin^{-1}\left(\frac{x-4}{5}\right)+C$
26.

$\tan^{-1}\left(\frac{x+3}{5}\right)+C$
27.

$\sqrt{x^{2}-11}-\sqrt{11}\sec^{-1}(x/\sqrt{11})+C$
28.

$\sqrt{x^{2}-3}+C$ (Trig. Subst. is not needed)
29.

$-\frac{1}{\sqrt{x^{2}+9}}+C$ (Trig. Subst. is not needed)
30.

$\frac{5}{2}x\sqrt{x^{2}-10}+25\ln\left\lvert\frac{x}{\sqrt{10}}+\frac{\sqrt{x^% {2}-10}}{\sqrt{10}}\right\rvert+C$
31.

$\frac{1}{18}\frac{x+2}{x^{2}+4x+13}+\frac{1}{54}\tan^{-1}\left(\frac{x+2}{3}% \right)+C$
32.

$\frac{x}{\sqrt{1-x^{2}}}-\sin^{-1}x+C$
33.

$\frac{1}{7}\left(-\frac{\sqrt{5-x^{2}}}{x}-\sin^{-1}(x/\sqrt{5})\right)+C$
34.

$\frac{1}{2}x\sqrt{x^{2}+3}-\frac{3}{2}\ln\left\lvert\frac{\sqrt{x^{2}+3}}{% \sqrt{3}}+\frac{x}{\sqrt{3}}\right\rvert+C$
35.

$\pi/2$
36.

$16\sqrt{3}-8\ln(2+\sqrt{3})$
37.

$2\sqrt{2}+2\ln(1+\sqrt{2})$
38.

$\pi/4+1/2)$
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 $\cos\big(\sin^{-1}(1/3)\big)=\cos\big(\sin^{-1}(-1/3)\big)=2\sqrt{2}/3$ .
40.

$\pi/8$
41.
- $\pi(1-\frac{\pi}{4})$
  
  $\pi(\sqrt{2}-\ln(1+\sqrt{2}))$

Exercises H.4

1.

rational
2.

T
3.

$\frac{A}{x}+\frac{B}{x-3}$
4.

$\frac{A}{x-3}+\frac{B}{x+3}$
5.

$\frac{A}{x-\sqrt{7}}+\frac{B}{x+\sqrt{7}}$
6.

$\frac{A}{x}+\frac{Bx+C}{x^{2}+7}$
7.

$3\ln\left\lvert x-2\right\rvert+4\ln\left\lvert x+5\right\rvert+C$
8.

$9\ln\left\lvert x+1\right\rvert-2\ln\left\lvert x\right\rvert+C$
9.

$\frac{1}{3}(\ln\left\lvert x+2\right\rvert-\ln\left\lvert x-2\right\rvert)+C$
10.

$\ln\left\lvert x+5\right\rvert-\frac{2}{x+5}+C$
11.

$-\frac{4}{x+8}-3\ln\left\lvert x+8\right\rvert+C$
12.

$\frac{5}{x+1}+7\ln\left\lvert x\right\rvert+2\ln\left\lvert x+1\right\rvert+C$
13.

$-\ln\left\lvert 2x-3\right\rvert+5\ln\left\lvert x-1\right\rvert+2\ln\left% \lvert x+3\right\rvert+C$
14.

$-\frac{1}{5}\ln\left\lvert 5x-1\right\rvert+\frac{2}{3}\ln\left\lvert 3x-1% \right\rvert+\frac{3}{7}\ln\left\lvert 7x+3\right\rvert+C$
15.

$x+\ln\left\lvert x-1\right\rvert-\ln\left\lvert x+2\right\rvert+C$
16.

$\frac{x^{2}}{2}+x+\frac{125}{9}\ln\left\lvert x-5\right\rvert+\frac{64}{9}\ln% \left\lvert x+4\right\rvert+C$
17.

$2x+C$
18.

$\frac{1}{6}\left(-\ln\left\lvert x^{2}+2x+3\right\rvert+2\ln\left\lvert x% \right\rvert-\sqrt{2}\tan^{-1}\left(\frac{x+1}{\sqrt{2}}\right)\right)+C$
19.

$\frac{1}{x}+\frac{1}{2}\ln\left\lvert\frac{x-1}{x+1}\right\rvert+C$
20.

$-\frac{3}{2}\ln\left\lvert x^{2}+4x+10\right\rvert+x+\frac{\tan^{-1}\left(% \frac{x+2}{\sqrt{6}}\right)}{\sqrt{6}}+C$
21.

$\ln\left\lvert 3x^{2}+5x-1\right\rvert+2\ln\left\lvert x+1\right\rvert+C$
22.

$2\ln\left\lvert x-3\right\rvert+2\ln\left\lvert x^{2}+6x+10\right\rvert-4\tan^% {-1}(x+3)+C$
23.

$\ln\left\lvert x\right\rvert-\frac{1}{2}\ln(x^{2}+1)-\tan^{-1}x-\frac{1}{2(x^{% 2}+1)}+C$
24.

$\frac{9}{10}\ln\left\lvert x^{2}+9\right\rvert+\frac{1}{5}\ln\left\lvert x+1% \right\rvert-\frac{4}{15}\tan^{-1}\left(\frac{x}{3}\right)+C$
25.

$\frac{1}{2}\left(3\ln\left\lvert x^{2}+2x+17\right\rvert-4\ln\left\lvert x-7% \right\rvert+\tan^{-1}\left(\frac{x+1}{4}\right)\right)+C$
26.

$\frac{1}{2}\ln(x^{2}+1)-\frac{1}{\sqrt{2}}\tan^{-1}\left(\frac{x}{\sqrt{2}}% \right)+C$
27.

$-\frac{1}{4}\ln(x^{2}+3)+\frac{1}{4}\ln(x^{2}+1)+C=\frac{1}{4}\ln\frac{x^{2}+1% }{x^{2}+3}+C$
28.

$\frac{-1}{2(x^{2}+2x+4)}-\frac{2\sqrt{3}}{9}\tan^{-1}\left(\frac{x+1}{\sqrt{3}% }\right)-\frac{2(x+1)}{3(x^{2}+2x+4)}+C$
29.

$3\left(\ln\left\lvert x^{2}-2x+11\right\rvert+\ln\left\lvert x-9\right\rvert% \right)+3\sqrt{\frac{2}{5}}\tan^{-1}\left(\frac{x-1}{\sqrt{10}}\right)+C$
30.

$\frac{1}{2}\ln\left\lvert x^{2}+10x+27\right\rvert+5\ln\left\lvert x+2\right% \rvert-6\sqrt{2}\tan^{-1}\left(\frac{x+5}{\sqrt{2}}\right)+C$
31.

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

$\ln x-\ln\left\lvert x+1\right\rvert+C$
33.

$\ln x-\frac{1}{2}\ln(x^{2}+1)+\frac{1}{2}\frac{1}{x^{2}+1}+C$
34.

$\tan^{-1}x-\frac{x}{x^{2}+1}+C$
35.

$\ln(2000/243)\approx 2.108$
36.

$5\ln(9/4)-\frac{1}{3}\ln(17/2)\approx 3.3413$
37.

$-\pi/4+\tan^{-1}3-\ln(11/9)\approx 0.263$
38.

$1/8$
39.

Exercises H.5

1.

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

$\displaystyle\frac{1}{6}\sin^{3}2x-\frac{1}{10}\sin^{5}2x+C$
3.

$\displaystyle 18\ln\left\lvert x-2\right\rvert-9\ln\left\lvert x-1\right\rvert% -5\ln\left\lvert x-3\right\rvert+C$
4.

$\displaystyle\frac{1}{5}\sec^{5}x+C$
5.

$\displaystyle\frac{x}{25\sqrt{x^{2}+25}}+C$
6.

$\displaystyle 2\ln\left\lvert\frac{2-\sqrt{4-x^{2}}}{x}\right\rvert+\sqrt{4-x^% {2}}+C$
7.

$\displaystyle 2\ln\left\lvert x-1\right\rvert-\ln\left\lvert x\right\rvert-% \frac{1}{x-1}-\frac{1}{(x-1)^{2}}+C$
8.

$\displaystyle-\sqrt{4+4x-x^{2}}+2\sin^{-1}\bigl(\frac{x-2}{\sqrt{8}}\bigr)+C$
9.

$\displaystyle\frac{1}{2}e^{x^{2}}(x^{2}-1)+C$
10.

$\displaystyle 3\sqrt[3]{x+8}+\ln[\sqrt[3]{x+8}-2]^{2}-\ln\left\lvert\sqrt[3]{(% x+8)^{2}}+2\sqrt[3]{x+8}+4\right\rvert-\frac{6}{\sqrt{3}}\tan^{-1}\frac{\sqrt[% 3]{x+8}+1}{\sqrt{3}}+C$
11.

$\displaystyle\frac{1}{13}e^{2x}(2\sin 3x-3\cos 3x)+C$
12.

$\displaystyle\frac{1}{4}\sin^{4}x-\frac{1}{6}\sin^{6}x+C$
13.

$\displaystyle-\sqrt{4-x^{2}}+C$
14.

$\displaystyle\frac{1}{3}x^{3}-x^{2}+3x-\frac{1}{2x}-\frac{1}{4}\ln\left\lvert x% \right\rvert-\frac{23}{4}\ln\left\lvert x+2\right\rvert+C$
15.

$\displaystyle 2\tan^{-1}\sqrt{x}+C$
16.

$\displaystyle\ln\left\lvert\sec e^{x}+\tan e^{x}\right\rvert+C$
17.

$\displaystyle\frac{1}{27}[6x\sin 3x-(9x^{2}-2)\cos 3x]+C$
18.

$\displaystyle\frac{2}{7}\cos^{7/2}x-\frac{2}{3}\cos^{3/2}x+C$
19.

$\displaystyle\frac{2}{3}(1+e^{x})^{3/2}+C$
20.

$\displaystyle\frac{1}{16}[2x\sqrt{4x^{2}+9}-9\ln(2x+\sqrt{4x^{2}+9}]+C$
21.

$\displaystyle\frac{1}{3}\tan^{3}x+C$
22.

$\displaystyle-x\csc x+\ln\left\lvert\csc x-\cot x\right\rvert+C$
23.

$\displaystyle-\frac{1}{4}(8-x^{3})^{4/3}+C$
24.

$\displaystyle 2\sin\sqrt{x}-2\sqrt{x}\cos\sqrt{x}+C$
25.

$\displaystyle\frac{1}{10}(3-2x)^{5/2}-\frac{1}{2}(3-2x)^{3/2}+C$
26.

$\displaystyle\frac{1}{2}e^{2x}-e^{x}+\ln(1+e^{x})+C$
27.

$\displaystyle\frac{2}{5}x^{5/2}-\frac{8}{3}x^{3/2}+6x^{1/2}+C$
28.

$\displaystyle\frac{1}{3}(16-x^{2})^{3/2}-16(16-x^{2})^{1/2}+C$
29.

$\displaystyle\frac{11}{2}\ln\left\lvert x+5\right\rvert-\frac{15}{2}\ln\left% \lvert x+7\right\rvert+C$
30.

$\displaystyle x\tan^{-1}5x-\frac{1}{10}\ln(1+25x^{2})+C$
31.

$\displaystyle e^{\tan x}+C$
32.

$\displaystyle\frac{1}{\sqrt{5}}\ln\left\lvert\sqrt{5}x+\sqrt{7+5x^{2}}\right% \rvert+C$
33.

$\displaystyle-\frac{1}{5}\cot^{5}x+\frac{1}{3}\cot^{3}x-\cot x-x+C$
34.

$\displaystyle\frac{1}{5}(x^{2}-25)^{5/2}+\frac{25}{3}(x^{2}-25)^{3/2}+C$
35.

$\displaystyle\frac{1}{3}x^{3}-\frac{1}{4}\tanh 4x+C$
36.

$\displaystyle-\frac{1}{4}x^{2}e^{-4x}-\frac{1}{8}xe^{-4x}-\frac{1}{32}e^{-4x}+C$
37.

$\displaystyle 3\sin^{-1}\bigl(\frac{x+5}{6}\bigr)+C$
38.

$\displaystyle\ln\frac{(x+3)^{2}(x^{2}+9)^{2}}{\left\lvert x-3\right\rvert^{5}}% +\frac{1}{3}\tan^{-1}\frac{x}{3}+C$
39.

$\displaystyle\frac{1}{3}\sec^{3}x-\sec x+C$
40.

$\displaystyle x^{3}\sin x+3x^{2}\cos x-6x\sin x-6\cos x+\sin x+C$
41.

$\displaystyle-2\sin^{-1}\bigl(\frac{2x}{3}\bigr)-\frac{1}{x}\sqrt{9-4x^{2}}+C$
42.

$\displaystyle 24x-\frac{10}{3}\ln\left\lvert\sin 3x\right\rvert-\frac{1}{3}% \cot 3x+C$
43.

$\displaystyle-\ln x+\frac{4}{\sqrt[4]{x}}+4\ln\left\lvert 1-\sqrt[4]{x}\right% \rvert+C$
44.

$\displaystyle-2\sqrt{1+\cos x}+C$
45.

$\displaystyle\frac{-x}{2(25+x^{2})}+\frac{1}{10}\tan^{-1}\bigl(\frac{x}{5}% \bigr)+C$
46.

$\displaystyle\ln(x^{2}+4)-\frac{3}{2}\tan^{-1}\frac{x}{2}+\frac{7}{\sqrt{5}}% \tan^{-1}\frac{x}{\sqrt{5}}+C$
47.

$\displaystyle\frac{1}{4}x^{4}-2x^{2}+4\ln\left\lvert x\right\rvert+C$
48.

$\displaystyle\frac{2}{5}x^{5/2}\ln x-\frac{4}{25}x^{5/2}+C$
49.

$\displaystyle\frac{3}{64}(2x+3)^{8/3}-\frac{9}{20}(2x+3)^{5/3}+\frac{27}{16}(2% x+3)^{2/3}+C$
50.

$\displaystyle\frac{e^{x}}{x+1}+C$
51.

$\displaystyle-\frac{1}{7}\cos 7x+C$
52.

$\displaystyle\frac{x^{2}}{2}\sin^{-1}x-\frac{1}{4}\sin^{-1}x+\frac{x}{4}\sqrt{% 1-x^{2}}+C$
53.
54.

$\ln\left\lvert 1+\tan\frac{\theta}{2}\right\rvert-\ln\left\lvert 1-\tan\frac{% \theta}{2}\right\rvert+C$
55.

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

Exercises H.6

1.

The interval of integration is finite, and the integrand is continuous on that interval.
2.

converge
3.

converges; could also state $\leq 10$ .
4.

$p>1$
5.

$p>1$
6.

$p<1$
7.

$e^{5}/2$
8.

$1/2$
9.

$1/3$
10.

$\pi/3$
11.

$1/\ln 2$
12.

diverges
13.

diverges
14.

diverges
15.

$1$
16.

diverges
17.

diverges
18.

diverges
19.

diverges
20.

diverges
21.

$2\sqrt{3}$
22.

$6$
23.

diverges
24.

diverges
25.

diverges
26.

$2+2\sqrt{2}$
27.

$1$
28.

$1/2$
29.

$0$
30.

$\pi/2$
31.

$-1/4$
32.

diverges
33.

$-1$
34.

$1$
35.

diverges
36.

$1/2$
37.

diverges; Limit Comparison Test with $1/x$ .
38.

converges; Limit Comparison Test with $1/x^{3/2}$ .
39.

diverges; Limit Comparison Test with $1/x$ .
40.

converges; Direct Comparison Test with $xe^{-x}$ .
41.

converges; Direct Comparison Test with $e^{-x}$ .
42.

converges; Direct Comparison Test with $xe^{-x}$ .
43.

converges; Direct Comparison Test with $1/(x^{2}-1)$ .
44.

diverges; Direct Comparison Test with $x/(x^{2}+1)$ .
45.

converges; Direct Comparison Test with $1/e^{x}$ .
46.

converges; Limit Comparison Test with $1/e^{x}$ .
47.
- $e^{-\lambda a}$
  
  $\frac{1}{\lambda}$
  
  $e^{-1}$
48.

Exercises H.7

1.

F
2.

When the antiderivative cannot be computed and when the integrand is unknown.
3.

They are superseded by the Trapezoidal Rule; it takes an equal amount of work and is generally more accurate.
4.

It is superseded by the Trapezoidal Rule; it is about as accurate, but takes more work.
5.
- $3/4$
  
  $2/3$
  
  $2/3$
6.
- $250$
  
  $250$
  
  $250$
7.
- $\frac{1}{4}(1+\sqrt{2})\pi\approx 1.896$
  
  $\frac{1}{6}(1+2\sqrt{2})\pi\approx 2.005$
  
  $2$
8.
- $2+\sqrt{2}+\sqrt{3}\approx 5.15$
  
  $2/3(3+\sqrt{2}+2\sqrt{3})\approx 5.25$
  
  $16/3\approx 5.33$
9.
- $38.5781$
  
  $147/4\approx 36.75$
  
  $147/4\approx 36.75$
10.
- $0.2207$
  
  $0.2005$
  
  $1/5$
11.
- $0$
  
  $0$
  
  $0$
12.
- $9(1+\sqrt{3})/2\approx 12.294$
  
  $3+6\sqrt{3}\approx 13.392$
  
  $9\pi/2\approx 14.137$
13.

Trapezoidal Rule: $0.9006$

Simpson’s Rule: $0.90452$
14.

Trapezoidal Rule: $3.0241$

Simpson’s Rule: $2.9315$
15.

Trapezoidal Rule: $13.9604$

Simpson’s Rule: $13.9066$
16.

Trapezoidal Rule: $3.0695$

Simpson’s Rule: $3.14295$
17.

Trapezoidal Rule: $1.1703$

Simpson’s Rule: $1.1873$
18.

Trapezoidal Rule: $2.52971$

Simpson’s Rule: $2.5447$
19.

Trapezoidal Rule: $1.0803$

Simpson’s Rule: $1.077$
20.

Trapezoidal Rule: $3.5472$

Simpson’s Rule: $3.6133$
21.
- $n=161$ (using $\max\big(f\,^{\prime\prime}(x)\big)=1$ )
  
  $n=12$ (using $\max\big(f\,^{(4)}(x)\big)=1$ )
22.
- $n=150$ (using $\max\big(f\,^{\prime\prime}(x)\big)=1$ )
  
  $n=18$ (using $\max\big(f\,^{(4)}(x)\big)=7$ )
23.
- $n=1004$ (using $\max\big(f\,^{\prime\prime}(x)\big)=39$ )
  
  $n=62$ (using $\max\big(f\,^{(4)}(x)\big)=800$ )
24.
- $n=5591$ (using $\max\big(f\,^{\prime\prime}(x)\big)=300$ )
  
  $n=46$ (using $\max\big(f\,^{(4)}(x)\big)=24$ )
25.
- Area is $30.8667$ cm².
  
  Area is $308,667$ yd².
26.
- Area is 25.0667 cm²
  
  Area is 250,667 yd²
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 $\displaystyle\int_{x_{1}}^{x_{1}+2\Delta x}a(x-x_{1})^{2}+b(x-x_{1})+c% \operatorname{d}\!x=\frac{a(2\Delta x)^{3}}{3}+\frac{b(2\Delta x)^{2}}{2}+c(2% \Delta x)=\frac{4(y_{1}-2y_{2}+y_{3})\Delta x}{3}+(4y_{2}-y_{3}-3y_{1})\Delta x% +2y_{1}\Delta x=\frac{\Delta x}{3}(4y_{1}-8y_{2}+4y_{3}+12y_{2}-3y_{3}-9y_{1}+% 6y_{1})=\frac{\Delta x}{3}(y_{1}+4y_{2}+y_{3})$ .