$\int \frac{\mathrm{dx}}{{\mathrm{x}}^2 +\hspace{0.17em}4\mathrm{x} + 13}$ is equal to

• $\log \left({\mathrm{x}}^2 + 4\mathrm{x} + 13\right) +\hspace{0.17em}\mathrm{c}$

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

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

• $\frac{2\mathrm{x} +\hspace{0.17em}4}{{\left({\mathrm{x}}^2 + 4\mathrm{x} + 13\right)}^2} + \mathrm{c}$

722.

The value of ${\int }_2^3\frac{\mathrm{x} + 1}{{\mathrm{x}}^2\left(\mathrm{x} - 1\right)}\mathrm{dx}$ is

• $\log \left(\frac{16}{9}\right) + \frac{1}{6}$

• $\log \left(\frac{16}{9}\right) - \frac{1}{6}$

• $2\log \left(2\right) - \frac{1}{6}$

• $\log \left(\frac{4}{3}\right) - \frac{1}{6}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)\right)\mathrm{dx} + {\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)\right)\mathrm{dx} + {\int }_{2\mathrm{\pi }}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)\right)\mathrm{dx}$

is equal to

• $\sqrt{2} - 2$

• $2\sqrt{2} - 2$

• $3\sqrt{2} - 2$

• $4\sqrt{2} - 2$

$\int \frac{{\mathrm{a}}^{{\displaystyle \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\sqrt{{\mathrm{a}}^{- \mathrm{x}} - {\mathrm{a}}^{\mathrm{x}}}}\mathrm{dx}$ is equal to

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

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

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

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

The value of ${\int }_0^1\frac{{\mathrm{x}}^4 + 1}{{\mathrm{x}}^2 +\hspace{0.17em}1}d\mathrm{x}$ is

• $\frac{1}{6}\left(3 - 4\mathrm{\pi }\right)$

• $\frac{1}{6}\left(3\mathrm{\pi } + 4\right)$

• $\frac{1}{6}\left(3 + 4\mathrm{\pi }\right)$

• $\frac{1}{6}\left(3\mathrm{\pi } - 4\right)$

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

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

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

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

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

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

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

• $\log \left(\frac{17}{2}\right)$

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