The function f(x) = ${\int }_0^{\mathrm{x}}\log \left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right)\mathrm{dx}$ is

• an even function

• an odd function

• a periodic function

• None of these

$\int \frac{\sin \left(2\mathrm{x}\right)}{1 + {\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ =

• $- \frac{1}{2}\log \left(1 + {\cos }^2\left(\mathrm{x}\right)\right) + \mathrm{c}$

• $2\log \left(1 + {\cos }^2\left(\mathrm{x}\right)\right) + \mathrm{c}$

• $\frac{1}{2}\log \left(1 + \cos \left(2\mathrm{x}\right)\right) + \mathrm{c}$

• $- \log \left(1 + {\cos }^2\left(\mathrm{x}\right)\right) + \mathrm{c}$

$\int \frac{{\mathrm{e}}^{\mathrm{x}}\left(1 + \sin \left(\mathrm{x}\right)\right)}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ =

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{1 + \sin \left(\mathrm{x}\right)}{1 - \cos \left(\mathrm{x}\right)}\right) + \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\mathrm{x}\right) + \mathrm{c}$

$\int \frac{1 + \tan \left(\mathrm{x}\right)}{{\mathrm{e}}^{- \mathrm{x}}\cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• ${\mathrm{e}}^{- \mathrm{x}}\tan \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• ${\mathrm{e}}^{- \mathrm{x}}\sec \left(\mathrm{x}\right) + \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\sec \left(\mathrm{x}\right) + \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\mathrm{x}\right) + \mathrm{c}$

475.

${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\csc }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• - 1

• 1

• 0

• $\frac{1}{2}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\log \left(1 + \tan \left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{8}{\log }_{\mathrm{e}}\left(2\right)$

• $\frac{\mathrm{\pi }}{4}{\log }_2\left(\mathrm{e}\right)$

• $\frac{\mathrm{\pi }}{4}{\log }_{\mathrm{e}}\left(2\right)$

• $\frac{\mathrm{\pi }}{8}{\log }_{\mathrm{e}}\left(\frac{1}{2}\right)$

$\int \frac{\left({\mathrm{x}}^3 +\hspace{0.17em}3{\mathrm{x}}^2 + 3\mathrm{x} + 1\right)}{{\left(\mathrm{x} + 1\right)}^5}\mathrm{dx}$ is equal to

• $- \frac{1}{\left(\mathrm{x} + 1\right)} +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{5}\log \left(\mathrm{x} +\hspace{0.17em}1\right) + \mathrm{c}$

• $\log \left(\mathrm{x} + 1\right) +\hspace{0.17em}\mathrm{c}$

• ${\tan }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

$\int \frac{\csc \left(\mathrm{x}\right)}{{\cos }^2\left(1 + \log \left(\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right)\right)\right)}\mathrm{dx}$ is equal to :

• ${\sin }^2\left(1 + \log \left(\tan \left(\frac{\mathrm{x}}{2}\right)\right)\right) + \mathrm{c}$

• $\tan \left(1 + \log \left(\tan \left(\frac{\mathrm{x}}{2}\right)\right)\right) + \mathrm{c}$

• ${\sec }^2\left(1 + \log \left(\tan \left(\frac{\mathrm{x}}{2}\right)\right)\right) + \mathrm{c}$

• $- \tan \left(1 + \log \left(\tan \left(\frac{\mathrm{x}}{2}\right)\right)\right) + \mathrm{c}$

$\int \frac{\mathrm{dx}}{\mathrm{x}\sqrt{{\mathrm{x}}^6 - 16}}$ is equa to :

• $\frac{1}{3}{\sec }^{-1}\left(\frac{{\mathrm{x}}^3}{4}\right) +\hspace{0.17em}\mathrm{c}$

• ${\cosh }^{-1}\left(\frac{{\mathrm{x}}^3}{4}\right) +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{12}{\sec }^{-1}\left(\frac{{\mathrm{x}}^3}{4}\right) +\hspace{0.17em}\mathrm{c}$

• ${\sec }^{-1}\left(\frac{{\mathrm{x}}^3}{4}\right) +\hspace{0.17em}\mathrm{c}$

If I₁ = ${\mathrm{I}}_1 = {\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(\mathrm{x}\right)\mathrm{dx} \mathrm{and} {\mathrm{I}}_2 = {\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{xcos}\left(\mathrm{x}\right)\mathrm{dx}$, then which one of the following is true ?

• ${\mathrm{I}}_1 + {\mathrm{I}}_{2 }= \frac{\mathrm{\pi }}{2}$

• ${\mathrm{I}}_2 - {\mathrm{I}}_1= \frac{\mathrm{\pi }}{2}$

• ${\mathrm{I}}_1 + {\mathrm{I}}_{2 }= 0$

• ${\mathrm{I}}_1 = {\mathrm{I}}_{2 }$