$\int \frac{{\mathrm{x}}^3\sin \left[{\tan }^{-1}\left({\mathrm{x}}^4\right)\right]}{1 + {\mathrm{x}}^8}\mathrm{dx}$ is equal to :

• $\frac{1}{4}\cos \left[{\tan }^{-1}\left({\mathrm{x}}^4\right)\right] + \mathrm{c}$

• $\frac{1}{4}\sin \left[{\tan }^{-1}\left({\mathrm{x}}^4\right)\right] + \mathrm{c}$

• $- \frac{1}{4}\cos \left[{\tan }^{-1}\left({\mathrm{x}}^4\right)\right] + \mathrm{c}$

• $\frac{1}{4}{\sec }^{-1}\left[{\tan }^{-1}\left({\mathrm{x}}^4\right)\right] + \mathrm{c}$

I_n = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\tan }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$, then $\underset{\mathrm{n}\to \infty }{\lim }\mathrm{n}\left[{\mathrm{I}}_{\mathrm{n}} + {\mathrm{I}}_{\mathrm{n} +\hspace{0.17em}2}\right]$ equals :

• $\frac{1}{2}$

• 1

• $\infty$

• zero

303.

If $\int \mathrm{xf}\left(\mathrm{x}\right)\mathrm{dx} = \frac{\mathrm{f}\left(\mathrm{x}\right)}{2}$, then f(x) is equal to :

• e^x

• e^{- x}

• log(x)

• $\frac{{\mathrm{e}}^{{\mathrm{x}}^2}}{2}$

${\int }_0^2\left[{\mathrm{x}}^2\right]\mathrm{dx}$ is :

• $2 - \sqrt{2}$

• $2 + \sqrt{2}$

• $\sqrt{2} - 1$

• $- \sqrt{2} - \sqrt{3} + 5$

${\int }_0^{\mathrm{\pi }}\left|\cos \left(\mathrm{x}\right)\right|\mathrm{dx}$ is equal to :

• $\frac{1}{2}$

• - 2

• 1

• - 1

$\int \left(\frac{\sin \left(2\mathrm{x}\right)}{\sin \left(3\mathrm{x}\right)\sin \left(5\mathrm{x}\right)}\right)\mathrm{dx}$ is equal to :

• $\frac{1}{5}{\log }_{\mathrm{e}}\left|\sin \left(5\mathrm{x}\right)\right| - \frac{1}{3}{\log }_{\mathrm{e}}\left|\sin \left(3\mathrm{x}\right)\right| + \mathrm{c}$

• $\frac{1}{3}{\log }_{\mathrm{e}}\left|\sin \left(3\mathrm{x}\right)\right| - \frac{1}{5}{\log }_{\mathrm{e}}\left|\sin \left(5\mathrm{x}\right)\right|$

• $\frac{1}{3}{\log }_{\mathrm{e}}\left|\sin \left(3\mathrm{x}\right)\right| + \frac{1}{5}{\log }_{\mathrm{e}}\left|\sin \left(5\mathrm{x}\right)\right|$

• $- \frac{1}{2}\cos \left(2\mathrm{x}\right) + \frac{1}{3}{\log }_{\mathrm{e}}\left|\sin \left(3\mathrm{x}\right)\right|$

$\int {\mathrm{e}}^{\mathrm{x}}\left\{\log \left(\sin \left(\mathrm{x}\right)\right) + \cot \left(\mathrm{x}\right)\right\}\mathrm{dx}$ is equal to

• e^xcot(x) + c

• e^xlog(sin(x)) + c

• e^xlog(sin(x)) + tan(x) + c

• e^x + sin(x) + c

${\int }_{- 10}^{10}\log \left(\frac{\mathrm{a} + \mathrm{x}}{\mathrm{a} - \mathrm{x}}\right)\mathrm{dx}$ is equal to :

• 0

• - 2log(a + 10)

• $2\log \left(\frac{\mathrm{a} + 10}{\mathrm{a} - 10}\right)$

• 2log(a + 10)

Define f(x) = ${\int }_0^{\mathrm{x}}\sin \left(\mathrm{t}\right)\mathrm{dt}, \mathrm{x} \ge 0$, Then :

• f is increasing only in the interval $\left[0, \frac{\mathrm{\pi }}{2}\right]$

• f is decreasing in the interval $\left[0, \mathrm{\pi }\right]$

• f attains maximum at x = $\frac{\mathrm{\pi }}{2}$

• f attains minimum at x = $\mathrm{\pi }$

Let f(x) = $\frac{{\sin }^2\left(\mathrm{\pi x}\right)}{1 + {\mathrm{\pi }}^2}$. Then, $\int \left(\mathrm{f}\left(\mathrm{x}\right) + \mathrm{f}\left(- \mathrm{x}\right)\right)\mathrm{dx}$ is equal to :

• 0

• x + c

• $\frac{\mathrm{x}}{2} - \frac{\cos \left(\mathrm{\pi x}\right)}{2\mathrm{\pi }} + \mathrm{c}$

• $\frac{\mathrm{x}}{2} - \frac{\sin \left(2\mathrm{\pi x}\right)}{4\mathrm{\pi }} + \mathrm{c}$