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Integrals

Multiple Choice Questions

221.

$\int \frac{(1 + x) e^{x}}{\sin^{2} ({xe}^{x})} dx$ is equal to

$- \cot (e^{x}) + C$
$\tan ({xe}^{x}) + C$
$\tan (e^{x}) + C$
$- \cot ({xe}^{x}) + C$

222.

$\int \frac{{xe}^{x}}{(1 + x^{2})} dx$ is equal to

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

223.

$\int e^{x} (\sin (x) + 2 \cos (x)) \sin (x) dx$ is equal to

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

224.

$\int \sqrt{1 + \cos (x)} dx$ is equal to

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

225.

$\int \frac{\sqrt{x^{2} - 1}}{x} dx$ is equal to

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

226.
$\int \frac{\sqrt{5 + x^{2}}}{x^{4}} dx$ is equal to

$\frac{1}{15} {(1 + \frac{5}{x^{2}})}^{\frac{3}{2}}$ + C

$\frac{- 1}{15} {(1 + \frac{1}{x^{2}})}^{\frac{3}{2}} + C$

$\frac{- 1}{15} {(1 + \frac{5}{x^{2}})}^{\frac{3}{2}}$

$\frac{1}{15} {(1 + \frac{1}{x^{2}})}^{\frac{3}{2}}$

$\frac{- 1}{15} {(1 + \frac{5}{x^{2}})}^{\frac{3}{2}}$

$Let I = \int \frac{\sqrt{5 + x^{2}}}{x^{4}} dx = \int \frac{x \sqrt{\frac{5}{x^{2}} + 1}}{x^{4}} dx = \int \frac{\sqrt{\frac{5}{x^{2}} + 1}}{x^{3}} dx Put \frac{5}{x^{2}} + 1 = t \Rightarrow - \frac{2 \times 5}{x^{3}} dx = dt \Rightarrow \frac{1}{x^{3}} dx = - \frac{1}{10} ∴ I = - \frac{1}{10} \int \sqrt{t} dt = - \frac{1}{10} . \frac{t^{\frac{3}{2}}}{\frac{3}{2}} + C = - \frac{1}{15} t^{\frac{3}{2}} + C = - \frac{1}{15} {(\frac{5}{x^{2}} + 1)}^{\frac{3}{2}} + C$