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

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

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

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

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

592.

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

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

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

Let f(x) = x - [x], for every real x, where [x] is the greatest integer less than or equal to x. Then, ${\int }_{- 1}^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is :

• 1

• 2

• 3

• 0

If ${\int }_0^{{\mathrm{x}}^2}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt} = \mathrm{xcos}\left(\mathrm{\pi x}\right)$, then the value of f(4) is :

• 1

• $\frac{1}{4}$

• - 1

• $\frac{- 1}{4}$

If f(x) = $\frac{2 - \mathrm{xcos}\left(\mathrm{x}\right)}{2 + \mathrm{xcos}\left(\mathrm{x}\right)} \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)\hspace{0.17em}= {\log }_{\mathrm{e}}\left(\mathrm{x}\right), (\mathrm{x} > 0)$ then the value of integral ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{g}\left(\mathrm{f}\right(\mathrm{x}\left)\right)$dx is :

• log_e(3)

• log_e(1)

• log_e(2)

• log_ee

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

(where c is a constant of integration)

• $\mathrm{x} + 2\sin \left(\mathrm{x}\right) + 2\sin \left(2\mathrm{x}\right) + \mathrm{c}$

• $2\mathrm{x} + \sin \left(\mathrm{x}\right) + \sin \left(2\mathrm{x}\right) + \mathrm{c}$

• $2\mathrm{x} + \sin \left(2\mathrm{x}\right) + 2\sin \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{x} + \sin \left(2\mathrm{x}\right) + 2\sin \left(\mathrm{x}\right) + \mathrm{c}$

If $\int \frac{\mathrm{dx}}{{\mathrm{x}}^3{\left(1 + {\mathrm{x}}^6\right)}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}} = \mathrm{xf}\left(\mathrm{x}\right)(1 +\hspace{0.17em}{\mathrm{x}}^6{)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + \mathrm{C}$ where C is a constant of integration, then the function f(x) is equal to :

• - $\frac{1}{2{\mathrm{x}}^2}$

• $- \frac{1}{6{\mathrm{x}}^3}$

• $\frac{3}{{\mathrm{x}}^2}$

• $- \frac{1}{2{\mathrm{x}}^3}$