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

• e^x + C

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

• ${\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right) + \mathrm{C}$

• $\tan \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

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

• 4

• 2

• 0

• 1

523.

The value of $\int \frac{{\mathrm{e}}^{\mathrm{x}}\left(1 +\hspace{0.17em}\mathrm{x}\right)}{{\cos }^2\left({\mathrm{e}}^{\mathrm{x}} . \mathrm{x}\right)}\mathrm{dx}$ is equal to

• $- \cot \left({\mathrm{ex}}^{\mathrm{x}}\right)$ + C

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

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

• $\cot \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{C}$

The vate of $\int \frac{{\mathrm{e}}^{\mathrm{x}}\left({\mathrm{x}}^2{\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{x}\right) + 1\right)}{{\mathrm{x}}^2 + 1}\mathrm{dx}$ is equal to

• ${\mathrm{e}}^{\mathrm{x}}{\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

• ${\tan }^{-1}\left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{C}$

• ${\tan }^{-1}\left({\mathrm{x}}^{\mathrm{e}}\right) + \mathrm{C}$

• ${\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{x}\right)} + \mathrm{C}$

The value of ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\sin }^{103}\left(\mathrm{x}\right) . {\cos }^{101}\left(\mathrm{x}\right)\mathrm{dx}$ is

• ${\left(\frac{\mathrm{\pi }}{4}\right)}^{103}$

• ${\left(\frac{\mathrm{\pi }}{4}\right)}^{101}$

• 2

• 0

The value of $\int \frac{{\mathrm{e}}^{6\log \left(\mathrm{x}\right)} - {\mathrm{e}}^{5\log \left(\mathrm{x}\right)}}{{\mathrm{e}}^{4\log \left(\mathrm{x}\right)} - {\mathrm{e}}^{3\log \left(\mathrm{x}\right)}}\mathrm{dx}$ is equal to

• 0 + C

• $\frac{{\mathrm{x}}^3}{3} + \mathrm{C}$

• $\frac{3}{{\mathrm{x}}^3} + \mathrm{C}$

• $\frac{1}{\mathrm{x}} + \mathrm{C}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{{\sin }^{1000}\left(\mathrm{x}\right)}{{\sin }^{1000}\left(\mathrm{x}\right) + {\cos }^{1000}\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• 1000

• 1

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

The vaue of ${\int }_2^8\frac{\sqrt{10 - \mathrm{x}}}{\sqrt{\mathrm{x}} + \sqrt{10 - \mathrm{x}}}\mathrm{dx}$ is

• 10

• 0

• 8

• 3

${\int }_{\raisebox{1ex}{$- \mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\mathrm{dx}}{{\mathrm{e}}^{\sin \left(\mathrm{x}\right)} + 1}$

• 1

• 0

• $\frac{\mathrm{\pi }}{2}$

• $- \frac{\mathrm{\pi }}{2}$

${\int }_{- 5}^5\left|\mathrm{x} + 2\right|\mathrm{dx}$ is equal to

• 28

• 30

• 29

• 27