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Integrals

Multiple Choice Questions

801.

The value of the integral $\int_{- \frac{π}{4}}^{\frac{π}{4}} \log (\sec (θ) - \tan (θ)) dθ$ is

0
$\frac{π}{4}$
$π$
$\frac{π}{2}$

802.

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

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

803.

$\int \frac{1}{x^{2} {(x^{4} + 1)}^{\frac{3}{4}}} dx$ is equal to

$\frac{- {(1 + x^{4})}^{\frac{1}{4}}}{2 x} + C$
$\frac{- {(1 + x^{4})}^{\frac{1}{4}}}{x} + C$
$\frac{- {(1 + x^{4})}^{\frac{3}{4}}}{x} + C$
$\frac{- {(1 + x^{4})}^{\frac{1}{4}}}{x^{2}} + C$

804.
$\int_{0}^{\frac{π}{4}} \log (\frac{\sin (x) + \cos (x)}{\cos (x)}) dx$ is equal to

$\int_{0}^{\frac{π}{4}} \log (\frac{\sin (x) + \cos (x)}{\cos (x)}) dx$

$\frac{π}{4} \log (2)$

$\log (2)$

$\frac{π}{2} \log (2)$

$\int_{0}^{\frac{π}{4}} \log (\frac{\sin (x) + \cos (x)}{\cos (x)}) dx$

$I = \int_{0}^{\frac{π}{4}} \log (\frac{\sin (x) + \cos (x)}{\cos (x)}) dx \Rightarrow I = \int_{0}^{\frac{π}{4}} \log (1 + \tan (x)) dx . . . (i) = \int_{0}^{\frac{π}{4}} \log [1 + \tan (\frac{π}{4} - x)] dx = \int_{0}^{\frac{π}{4}} \log (1 + \frac{\tan (\frac{π}{4}) - \tan (x)}{1 + \tan (\frac{π}{4}) \tan (x)}) dx = \int_{0}^{\frac{π}{4}} \log (1 + \frac{1 - \tan (x)}{1 + \tan (x)}) dx \Rightarrow I = \int_{0}^{\frac{π}{4}} \log (\frac{2}{1 + \tan (x)}) dx . . . (ii) On adding Eqs (i) and (ii), we get 2 I = \int_{0}^{\frac{π}{4}} \log (2) dx = \log (2) (\frac{π}{4}) \Rightarrow I = \frac{π}{8} \log (2)$