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Integrals

Multiple Choice Questions

521.

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

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

522.

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

$\frac{1}{2} \log |\log (x^{2})| + C$
$\log |\log (x^{2})| + C$
$2 \log |\log (x^{2})| + C$
$\frac{1}{4} \log |\log (x^{2})| + C$

523.
$\int_{0}^{1} \frac{1}{(x^{2} + 16) (x^{2} + 25)} dx$ is equal to

$\frac{1}{5} [\frac{1}{4} \tan^{- 1} (\frac{1}{4}) - \frac{1}{5} \tan^{- 1} (\frac{1}{5})]$

$\frac{1}{9} [\frac{1}{4} \tan^{- 1} (\frac{1}{4}) - \frac{1}{5} \tan^{- 1} (\frac{1}{5})]$

$\frac{1}{4} [\frac{1}{4} \tan^{- 1} (\frac{1}{4}) - \frac{1}{5} \tan^{- 1} (\frac{1}{5})]$

$\frac{1}{9} [\frac{1}{5} \tan^{- 1} (\frac{1}{4}) - \frac{1}{4} \tan^{- 1} (\frac{1}{5})]$

$\frac{1}{9} [\frac{1}{4} \tan^{- 1} (\frac{1}{4}) - \frac{1}{5} \tan^{- 1} (\frac{1}{5})]$

$I = \int_{0}^{1} \frac{1}{(x^{2} + 16) (x^{2} + 25)} dx = \frac{1}{9} \int_{0}^{1} (\frac{1}{(x^{2} + 16)} - \frac{1}{(x^{2} + 25)}) dx$

$= \frac{1}{9} {[\frac{1}{4} \tan^{- 1} (\frac{x}{4}) - \frac{1}{5} \tan^{- 1} (\frac{x}{5})]}_{0}^{1} = \frac{1}{5} [\frac{1}{4} \tan^{- 1} (\frac{1}{4}) - \frac{1}{5} \tan^{- 1} (\frac{1}{5})]$

524.

$\int_{- 1}^{1} x (1 - x) (1 + x) dx$ is equal to

$\frac{1}{3}$
$\frac{2}{3}$
1
0

525.

The value of $\int_{- π}^{π} \frac{\sin^{2} (x)}{1 + 7^{x}} dx$ is

7^x
$π$
$\frac{π}{2}$
$2^{π}$

526.

$\int_{0}^{\frac{\sqrt{π}}{2}} 2 x^{3} \sin (x^{2}) dx$ is equal to

$\frac{1}{\sqrt{2}} (1 + \frac{π}{4})$
$\frac{1}{\sqrt{2}} (1 - \frac{π}{4})$
$\frac{1}{\sqrt{2}} (\frac{π}{2} - 1)$
$\frac{1}{\sqrt{2}} (1 - \frac{π}{2})$

527.

$\int {(\sec (x))}^{m} (\tan^{3} (x) + \tan (x)) dx$ is equal to

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

528.

$\int \frac{1}{7} \sin (\frac{x}{7} + 10) dx$ is equal to

$\frac{1}{7} \cos (\frac{x}{7} + 10) + C$
$- \frac{1}{7} \cos (\frac{x}{7} + 10) dx$
$- \cos (\frac{x}{7} + 10) + C$
$- 7 \cos (\frac{x}{7} + 10) + C$

529.

$\int (\frac{x - a}{x} - \frac{x}{x + a}) dx$ is equal to

$\log |\frac{x + a}{x}| + C$
$alog |\frac{x + a}{x}| + C$
$alog |\frac{x}{x + a}| + C$
$\log |\frac{x + a}{x}| + C$

530.

$\int x^{4} e^{x^{5}} \cos (e^{x^{5}}) dx$ is equal to

$\frac{1}{3} \sin (e^{x^{5}}) + C$
$\frac{1}{4} \sin (e^{x^{5}}) + C$
$\frac{1}{5} \sin (e^{x^{5}}) + C$
$\sin (e^{x^{5}}) + C$

Previous Year Papers

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Multiple Choice Questions

523. ∫011x2 + 16x2 + 25dx is equal to 1514tan-114 - 15tan-115 1914tan-114 - 15tan-115 1414tan-114 - 15tan-115 1915tan-114 - 14tan-115