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

• $- \mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• None of these

452.

$\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^5 + 1\right)}$ is equal to :

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

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

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

• None of these

$\int \frac{\mathrm{x} + \sin \left(\mathrm{x}\right)}{1 +\hspace{0.17em}\cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

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

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

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

• None of these

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}{\cos }^3\left(4\mathrm{\theta }\right)\mathrm{d\theta }$ is equal to

• $\frac{5}{3}$

• $\frac{5}{4}$

• $\frac{1}{3}$

• $\frac{1}{6}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right) \sin \left(\mathrm{x}\right)\hspace{0.17em}}\mathrm{dx}$ is equal to

• 0

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

$\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^7 + 1\right)}$ is equal to

• $\log \left(\frac{{\mathrm{x}}^7}{{\mathrm{x}}^7 + 1}\right) + \mathrm{c}$

• $\frac{1}{7}\log \left(\frac{{\mathrm{x}}^7}{{\mathrm{x}}^7 + 1}\right) + \mathrm{c}$

• $\log \left(\frac{{\mathrm{x}}^7 + 1}{{\mathrm{x}}^7}\right) + \mathrm{c}$

• $\frac{1}{7}\log \left(\frac{{\mathrm{x}}^7 + 1}{{\mathrm{x}}^7}\right) + \mathrm{c}$

${\int }_{- 1}^1\left|1 - \mathrm{x}\right|\mathrm{dx}$ is equal to

• - 2

• 0

• 2

• 4

$\int \sqrt{\mathrm{x}}{\mathrm{e}}^{\sqrt{\mathrm{x}}}\mathrm{dx}$ is equal to

• $2\sqrt{\mathrm{x}} - {\mathrm{e}}^{\sqrt{\mathrm{x}}} - 4\sqrt{{\mathrm{xe}}^{\sqrt{\mathrm{x}}}} +\hspace{0.17em}\mathrm{c}$

• $\left(2\sqrt{\mathrm{x}} - 4\sqrt{\mathrm{x}} + 4\right){\mathrm{e}}^{\sqrt{\mathrm{x}}} +\hspace{0.17em}\mathrm{c}$

• $\left(2\sqrt{\mathrm{x}} + 4\sqrt{\mathrm{x}} + 4\right){\mathrm{e}}^{\sqrt{\mathrm{x}}} +\hspace{0.17em}\mathrm{c}$

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

$\int \frac{\mathrm{dx}}{{\mathrm{x}}^2 +\hspace{0.17em}2\mathrm{x} + 2}$ is equal to

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

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

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

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

${\int }_0^{2\mathrm{\pi }}\left(\sin \left(\mathrm{x}\right) +\hspace{0.17em}\left|\sin \left(\mathrm{x}\right)\right|\right)\mathrm{dx}$ is equal to

• 4

• 0

• 1

• 8