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Integrals

Multiple Choice Questions

151.

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

$\frac{1}{8} (x^{2} - 1) + C$
$\frac{x^{2}}{4} + C$
$\frac{x}{2} + C$
$\frac{x^{2}}{2} + C$

152.

$\int_{\frac{π}{6}}^{\frac{π}{3}} \frac{1}{\tan^{3} (x)} dx$ is

$\frac{π}{12}$
$\frac{π}{4}$
$\frac{π}{3}$
$\frac{π}{6}$

153.

$\int_{- 1}^{1} \frac{17 x^{5} - x^{4} + 29 x^{3} - 31 x + 1}{x^{2} + 1} dx$ is

154.

If I_n = $\int_{0}^{\frac{π}{4}} \tan^{n} (x) dx$ , then $\frac{1}{I_{3} + I_{5}}$ is

155.
If $\int_{0}^{\frac{π}{2}} \sin^{6} (x) dx = \frac{5 π}{32}$ , then the value of $\int_{- π}^{π} (\sin^{6} (x) + \cos^{6} (x)) dx$ is

$\frac{5 π}{8}$

$\frac{5 π}{16}$

$\frac{5 π}{2}$

$\frac{5 π}{4}$

$\frac{5 π}{4}$

$Given that \int_{0}^{\frac{π}{2}} \sin^{6} (x) dx = \frac{5 π}{32} Let I = \int_{- π}^{π} (\sin^{6} (x) + \cos^{6} (x)) dx = 2 \int_{0}^{π} (\sin^{6} (x) + \cos^{6} (x)) dx = 4 \int_{0}^{\frac{π}{2}} (\sin^{6} (x) + \cos^{6} (x)) dx = 4 \int_{0}^{\frac{π}{2}} \sin^{6} (x) dx + 4 \int_{0}^{\frac{π}{2}} \cos (\frac{π}{2} - x) dx = 8 \int_{0}^{\frac{π}{2}} \sin^{6} (x) dx = 8 \times \frac{5 . 3 . 1}{6 . 4 . 2} \times \frac{π}{2} = \frac{5 π}{4}$