Indefinite Integration

1. A.
   $p = \dfrac{\sqrt{2}}{5}, q = \dfrac{1}{5}$
2. B.
   $p = \dfrac{\sqrt{2}}{3}, q = 3$
3. C.
   $p = \dfrac{\sqrt{2}}{5}, q = 5$
4. D.
   $p = \dfrac{2}{\sqrt{5}}, q = \sqrt{5}$

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1. A.
   $a$
2. B.
   $-1$
3. C.
   $-a$
4. D.
   $1$

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1. A.
   $-\dfrac{1}{16}$
2. B.
   $-\dfrac{25}{16}$
3. C.
   $\dfrac{25}{4}$
4. D.
   $-\dfrac{25}{4}$

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1. A.
   $\frac{e^{x}}{(x + 6)^{2}} + c$, where c is the constant of integration.
2. B.
   $\frac{e^{x}}{x + 5} + c$, where $c$ is the constant of integration.
3. C.
   $\frac{e^{x}}{(x + 5)^{2}} + c$, where c is the constant of integration.
4. D.
   $\frac{e^{x}}{x + 6} + c$, where c is the constant of integration.

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1. A.
   $e^x \cot x + C$
2. B.
   $e^x \log\sin x + C$
3. C.
   $e^x \log\sin x + \tan x + C$
4. D.
   $e^x + \sin x + C$

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1. A.
   $3\log|x^{5/2} + 4x^{3/2}| + c$
2. B.
   $\dfrac{1}{2}\log|x + 4| + c$
3. C.
   $4\sec(x^{3/2}) + \log\left(\tan\left(\dfrac{x}{2}\right)\right) + c$
4. D.
   $\tan^{-1}\left(\dfrac{\sqrt{x}}{2}\right) + c$

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1. A.
   $\dfrac{1}{1 + \log x} + c$
2. B.
   $-\dfrac{1}{1 + \log x} + c$
3. C.
   $\dfrac{x}{1 + \log x} + c$
4. D.
   $-\dfrac{x}{1 + \log x} + c$

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1. A.
   $-\!\left(\dfrac{x^4+1}{x^4}\right)^{1/4}+c$
2. B.
   $\left(\dfrac{x^4+1}{1}\right)^{1/4}+c$
3. C.
   $-\!\left(\dfrac{x^4+1}{1}\right)^{1/4}+c$
4. D.
   $-\!\left(\dfrac{x^4+1}{x^4}\right)^{1/4}+c$

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1. A.
   $-\dfrac{8}{5}\log|4 - x| - \dfrac{2}{5}\log|1 + x| + c$
2. B.
   $-\dfrac{2}{5}\log|4 + x| - \dfrac{8}{5}\log|1 - x| + c$
3. C.
   $-\dfrac{2}{5}\log|4 - x| - \dfrac{8}{5}\log|1 + x| + c$
4. D.
   $-\dfrac{8}{5}\log|4 + x| - \dfrac{2}{5}\log|1 - x| + c$

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1. A.
   $\dfrac{1}{\sqrt{13}}\tan^{-1}\left(\dfrac{4\tan x + 1}{\sqrt{13}}\right) + c$
2. B.
   $\dfrac{2}{\sqrt{11}}\tan^{-1}\left(\dfrac{4\tan x + 2}{\sqrt{11}}\right) + c$
3. C.
   $\dfrac{1}{11}\tan^{-1}\left(\dfrac{2\tan x + 1}{11}\right) + c$
4. D.
   $\dfrac{1}{\sqrt{11}}\tan^{-1}\left(\dfrac{4\tan x + 1}{\sqrt{11}}\right) + c$

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