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Integrals

Multiple Choice Questions

381.

The value of $\int \frac{dx}{x (x^{n} + 1)}$ is

$\frac{1}{n} \log (\frac{x^{n}}{x^{n} + 1}) + C$
$\log (\frac{x^{n} + 1}{x^{n}}) + C$
$\frac{1}{n} \log (\frac{x^{n} + 1}{x^{n}}) + C$
$\log (\frac{x^{n}}{x^{n} + 1}) + C$

382.

The value of $\int \cos (\log (x)) dx$ is

$\frac{1}{2} \{\sin (\log (x)) + \cos (\log (x))\} + C$
$\frac{x}{2} \{\sin (\log (x)) + \cos (\log (x))\} + C$
$\frac{x}{2} \{\sin (\log (x)) - \cos (\log (x))\} + C$
$\frac{1}{2} \{\sin (\log (x)) - \cos (\log (x))\} + C$

383.

The value of $\int e^{x} [\frac{1 + \sin (x)}{1 + \cos (x)}] dx$ is

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

384.
The value of $\int \frac{1}{3 \sin (x) - \cos (x) + 3} dx$ is

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

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

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

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

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

$Let I = \int \frac{dx}{3 \sin (x) - \cos (x) + 3} [∵ \sin (x) = \frac{2 \tan (\frac{x}{2})}{1 + \tan^{2} (\frac{x}{2})} and \cos (x) = \frac{1 - \tan^{2} (\frac{x}{2})}{1 + \tan (\frac{x}{2})}] I = \int \frac{dx}{3 \frac{2 \tan (\frac{x}{2})}{1 + \tan^{2} (\frac{x}{2})} - \frac{1 - \tan^{2} (\frac{x}{2})}{1 + \tan^{2} (\frac{x}{2})} + 3} = \int \frac{(1 + \tan^{2} (\frac{x}{2})) dx}{6 \tan (\frac{x}{2}) - 1 + \tan^{2} (\frac{x}{2}) + 3 + 3 \tan^{2} (\frac{x}{2})} = \int \frac{\sec^{2} (\frac{x}{2})}{4 \tan^{2} (\frac{x}{2}) + 6 \tan (\frac{x}{2}) + 2} dx [Let t = \tan (\frac{x}{2}), dt = \frac{1}{2} \sec^{2} (\frac{x}{2}) dx]$

$= \int \frac{dt}{2 t^{2} + 3 t + 1} = \int \frac{dt}{(t + 1) (2 t + 1)} = \int \{\frac{- 1}{(t + 1)} + \frac{2}{(2 t + 1)}\} dt [∵ by partial fraction] = - \log (t + 1) + \frac{2}{2} \log (2 t + 1) + C = \log (\frac{(2 t + 1)}{(t + 1)}) + C = \log (\frac{2 \tan (\frac{x}{2}) + 1}{\tan (\frac{x}{2}) + 1}) + C$

385.

The value of $\int \frac{\sin (2 x)}{\sin^{4} (x) + \cos^{4} (x)} dx$ is

$\tan^{- 1} (\cot^{2} (x)) + C$
$\tan^{- 1} (\cos (2 x)) + C$
$\tan^{- 1} (\sin (2 x)) + C$
$\tan^{- 1} (\tan^{2} (x)) + C$

386.

The value of $\int \sqrt{1 + \sec (x)} dx$ is

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

387.

The value of $\int \frac{(x^{2} + 1)}{x^{4} + x^{2} + 1} dx$ is

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