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Integrals

Multiple Choice Questions

521.

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

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

522.

$\int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{dx}{1 + \cos (2 x)}$ is equal to

523.

The value of $\int \frac{e^{x} (1 + x)}{\cos^{2} (e^{x} . x)} dx$ is equal to

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

524.

The vate of $\int \frac{e^{x} (x^{2} \tan^{- 1} (x) + \tan^{- 1} (x) + 1)}{x^{2} + 1} dx$ is equal to

$e^{x} \tan^{- 1} (x) + C$
$\tan^{- 1} (e^{x}) + C$
$\tan^{- 1} (x^{e}) + C$
$e^{\tan^{- 1} (x)} + C$

525.

The value of $\int_{- \frac{π}{4}}^{\frac{π}{4}} \sin^{103} (x) . \cos^{101} (x) dx$ is

${(\frac{π}{4})}^{103}$
${(\frac{π}{4})}^{101}$
2
0

526.

The value of $\int \frac{e^{6 \log (x)} - e^{5 \log (x)}}{e^{4 \log (x)} - e^{3 \log (x)}} dx$ is equal to

0 + C
$\frac{x^{3}}{3} + C$
$\frac{3}{x^{3}} + C$
$\frac{1}{x} + C$

527.
$\int_{0}^{\frac{π}{2}} \frac{\sin^{1000} (x)}{\sin^{1000} (x) + \cos^{1000} (x)} dx$ is equal to

1000

1

$\frac{π}{2}$

$\frac{π}{4}$

$\frac{π}{4}$

$Let I = \int_{0}^{\frac{π}{2}} \frac{\sin^{1000} (x)}{\sin^{1000} (x) + \cos^{1000} (x)} dx . . . (i) \Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{\sin^{1000} (\frac{π}{2} - x)}{\sin^{1000} (\frac{π}{2} - x) + \cos^{1000} (\frac{π}{2} - x)} dx [∵ \int_{0}^{a} f (x) dx = \int_{0}^{a} f (a - x) dx] \Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{\cos^{1000} (x)}{\sin^{1000} (x) + \cos^{1000} (x)} dx . . . (ii) On addingEqs . (i) and (ii), we get 2 I = \int_{0}^{\frac{π}{2}} \frac{\sin^{1000} (x) + \cos^{1000} (x)}{\sin^{1000} (x) + \cos^{1000} (x)} dx \Rightarrow 2 I = \int_{0}^{\frac{π}{2}} 1 dx = {[x]}_{0}^{\frac{π}{2}} \Rightarrow 2 I = \frac{π}{2} ∴ I = \frac{π}{4}$