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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

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

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

388.
The value of $\int_{0}^{1} x^{2} {(1 - x^{2})}^{\frac{3}{2}} dx$ is

$\frac{1}{32}$

$\frac{π}{8}$

$\frac{π}{16}$

$\frac{π}{32}$

$\frac{π}{32}$

$Let I = \int_{0}^{1} x^{2} {(1 - x^{2})}^{\frac{3}{2}} dx (let x = \sin (θ) \Rightarrow dx = \cos (θ) dθ) I = \int_{0}^{\frac{π}{2}} \sin^{2} (θ) {(\cos^{2} (θ))}^{\frac{3}{2}} . \cos (θ) dθ = \int_{0}^{\frac{π}{2}} \sin^{4} (θ) . \cos^{4} (θ) By Gamma function, I = \frac{Γ \frac{2 + 1}{2} Γ \frac{4 + 1}{2}}{2 Γ \frac{2 + 4 + 2}{2}} = \frac{Γ \frac{3}{2} Γ \frac{5}{2}}{2 Γ 4} = \frac{\frac{1}{2} Γ . \frac{3}{2} . \frac{1}{2} . Γ \frac{1}{2}}{2 . 3 . 2} = \frac{3 . \sqrt{π} . \sqrt{π}}{2 . 2 . 2 . 2 . 3 . 2} = \frac{π}{32}$