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

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

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

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

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

$\int 32 {\mathrm{x}}^3 {\left(\log \left(\mathrm{x}\right)\right)}^2\mathrm{dx}$ is equal to

• 8x⁴(log(x))² + c

• ${\mathrm{x}}^4\left\{8{\left(\log \left(\mathrm{x}\right)\right)}^2 - 4\left(\log \left(\mathrm{x}\right)\right) + 1\right\} + \mathrm{c}$

• $\left\{8{\left(\log \left(\mathrm{x}\right)\right)}^2 - 4\left(\log \left(\mathrm{x}\right)\right)\right\} + \mathrm{c}$

• $x^3\left\{8{\left(\log \left(\mathrm{x}\right)\right)}^2 - 2\left(\log \left(\mathrm{x}\right)\right)\right\} + \mathrm{c}$

443.

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

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

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

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

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

If $\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{g}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}, \mathrm{then} \int {\mathrm{f}}^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to :

• xf^-1(x) + c

• f(g^-1(x)) + c

• xf^-1(x) - g(f^-1(x)) + c

• g^-1(x) + c

The value of ${\int }_1^2\frac{\mathrm{dx}}{\mathrm{x}\left(1 + {\mathrm{x}}^4\right)}$ is :

• $\frac{1}{4}\log \left(\frac{17}{32}\right)$

• $\frac{1}{4}\log \left(\frac{32}{17}\right)$

• $\log \left(\frac{17}{2}\right)$

• $\frac{1}{4}\log \left(\frac{17}{2}\right)$

The value of the integral ${\int }_{\mathrm{a}}^{\mathrm{b}}\frac{\sqrt{\mathrm{x}}\mathrm{dx}}{\sqrt{\mathrm{x}} + \sqrt{\mathrm{a} + \mathrm{b} - \mathrm{x}}}$ is :

• $\mathrm{\pi }$

• $\frac{1}{2}\left(\mathrm{b} - \mathrm{a}\right)$

• $\mathrm{\pi }/2$

• b - a

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

• 1

• - 1

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

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

• $\frac{{\mathrm{\pi }}^2}{32}$

• $\frac{{\mathrm{\pi }}^2}{16}$

• $\frac{\mathrm{\pi }}{32}$

• None of these

${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{\mathrm{xsin}\left(\mathrm{x}\right)}{{\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is :

• $\frac{1}{3}\left(4\mathrm{\pi } + 1\right)$

• $\frac{4\mathrm{\pi }}{3} - 2\log \left(\tan \left(\frac{5\mathrm{\pi }}{12}\right)\right)$

• $\frac{4\mathrm{\pi }}{3} + \log \left(\tan \left(\frac{5\mathrm{\pi }}{12}\right)\right)$

• None of these

$\int \frac{{\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^3}{\left(1 + {\mathrm{x}}^2\right)}\mathrm{dx}$ is equa to :

• $3{\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^2 + \mathrm{c}$

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

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

• None of these