${\int }_0^{\mathrm{\pi }}\frac{\mathrm{xdx}}{{\mathrm{a}}^2{\cos }^2\left(\mathrm{x}\right) + {\mathrm{b}}^2{\sin }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{2\mathrm{ab}}$

• $\frac{\mathrm{\pi }}{\mathrm{ab}}$

• $\frac{{\mathrm{\pi }}^2}{2\mathrm{ab}}$

• $\frac{{\mathrm{\pi }}^2}{\mathrm{ab}}$

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

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

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

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

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

$\int \sqrt{1 + \sin \left(\frac{\mathrm{x}}{4}\right)}\mathrm{dx}$ is equal to

• $8\left(\sin \left(\frac{\mathrm{x}}{8}\right) + \cos \left(\frac{\mathrm{x}}{8}\right)\right) + \mathrm{C}$

• $8\left(\sin \left(\frac{\mathrm{x}}{8}\right) - \cos \left(\frac{\mathrm{x}}{8}\right)\right) + \mathrm{C}$

• $8\left(\cos \left(\frac{\mathrm{x}}{8}\right) - \sin \left(\frac{\mathrm{x}}{8}\right)\right) + \mathrm{C}$

• $\frac{1}{8}\left(\sin \left(\frac{\mathrm{x}}{8}\right) - \cos \left(\frac{\mathrm{x}}{8}\right)\right) + \mathrm{C}$

${\int }_0^{\infty }\frac{\mathrm{xdx}}{\left(1 + \mathrm{x}\right)\left(1 + {\mathrm{x}}^2\right)}$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• 0

• 1

• $\frac{\mathrm{\pi }}{4}$

If I_n = $\int {\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}}\mathrm{dx}$, then I_n + nI_{n - 1} is equal to

• ${\left(\mathrm{xlog}\left(\mathrm{x}\right)\right)}^{\mathrm{n}}$

• $\mathrm{x}{\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}}$

• $\mathrm{n}{\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}}$

• ${\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n} - 1}$

The value of $\int \frac{\mathrm{dx}}{\sqrt{2\mathrm{x} - {\mathrm{x}}^2}}$ is

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

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

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

• $- \sqrt{2\mathrm{x} - {\mathrm{x}}^2} + \mathrm{c}$

The value of $\int \mathrm{xlog}\left({\mathrm{x}}^3\right)\mathrm{dx}$ is

• $\frac{1}{16}\left(4{\mathrm{x}}^4\log \left(\mathrm{x}\right) - {\mathrm{x}}^4 + \mathrm{c}\right)$

• $\frac{1}{8}\left({\mathrm{x}}^4\log \left(\mathrm{x}\right) - 4{\mathrm{x}}^4 + \mathrm{c}\right)$

• $\frac{{\mathrm{x}}^4\log \left(\mathrm{x}\right)}{4} + \mathrm{c}$

• $\frac{1}{16}\left(4{\mathrm{x}}^4\log \left(\mathrm{x}\right) + {\mathrm{x}}^4 + \mathrm{c}\right)$

The value of ${\int }_0^{\mathrm{\pi }}\frac{\mathrm{dx}}{5 + 3\cos \left(\mathrm{x}\right)}$ is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{2}$

• zero

The value of ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\log \left(\tan \left(\mathrm{x}\right)\right)\mathrm{dx}$ is

• - 1

• $\frac{1}{2}$

• zero

• 1

The value of ${\int }_{- \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\cos \left(\mathrm{x}\right)\left[\log \left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right)\right]\mathrm{dx}$ is

• 2e^1/2

• 1

• e^1/2

• zero