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

• $\log \left|\cos \left({\mathrm{xe}}^{\mathrm{x}}\right)\right| + \mathrm{C}$

• $\log \left|\cot \left({\mathrm{xe}}^{\mathrm{x}}\right)\right| + \mathrm{C}$

• $\log \left|\sec \left({\mathrm{xe}}^{- \mathrm{x}}\right)\right| + \mathrm{C}$

• $\log \left|\sec \left({\mathrm{xe}}^{\mathrm{x}}\right)\right| + \mathrm{C}$

$\int \frac{{\mathrm{x}}^5}{\sqrt{1 + {\mathrm{x}}^3}}\mathrm{dx}$ is equal to

• $\frac{2}{9}\sqrt{1 + {\mathrm{x}}^2}\left({\mathrm{x}}^3 - 9\right)$ + C

• $\frac{2}{9}\sqrt{{\mathrm{x}}^3 - 9}\left(1 + {\mathrm{x}}^2\right)$ + C

• $\frac{2}{9}\sqrt{1 + {\mathrm{x}}^3}$ + C

• $\frac{2}{9}\sqrt{1 + {\mathrm{x}}^2}\left({\mathrm{x}}^3 - 2\right)$ + C

$\int \frac{4{\mathrm{e}}^{\mathrm{x}}}{2{\mathrm{e}}^{\mathrm{x}} - 5{\mathrm{e}}^{- \mathrm{x}}}\mathrm{dx}$ is equal to

• $4\log \left|{\mathrm{e}}^{\mathrm{x}} - 5\right| + \mathrm{C}$

• $\frac{1}{4}\log \left|{\mathrm{e}}^{2\mathrm{x}} - 5\right| + \mathrm{C}$

• $\log \left|2{\mathrm{e}}^{2\mathrm{x}} - 5{\mathrm{e}}^{- \mathrm{x}}\right| + \mathrm{C}$

• $\log \left|2{\mathrm{e}}^{2\mathrm{x}} - 5\right| + \mathrm{C}$

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

• $\frac{{\mathrm{x}}^2}{2} + 2\mathrm{x} + \log \left|\mathrm{x}\right| + \mathrm{C}$

• $\frac{{\mathrm{x}}^2}{2} + 2 + \log \left|\mathrm{x}\right| + \mathrm{C}$

• $\frac{{\mathrm{x}}^2}{2} + \mathrm{x} + \log \left|\mathrm{x}\right| + \mathrm{C}$

• $\frac{{\mathrm{x}}^2}{2} + 2\mathrm{x} + 2\log \left|\mathrm{x}\right| + \mathrm{C}$

$\int \frac{{\mathrm{x}}^{\mathrm{n} - 1}}{{\mathrm{x}}^{2\mathrm{n}} + {\mathrm{a}}^2}\mathrm{dx}$ is equal to

• $\frac{1}{\mathrm{na}}{\tan }^{-1}\left(\frac{{\mathrm{x}}^{\mathrm{n}}}{\mathrm{a}}\right) + \mathrm{C}$

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

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

• $\frac{\mathrm{n}}{\mathrm{a}}{\cot }^{-1}\left(\frac{{\mathrm{x}}^{\mathrm{n}}}{\mathrm{a}}\right) + \mathrm{C}$

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

• $\log \left|\mathrm{x}\left({\mathrm{x}}^2 + 1\right)\right| + \mathrm{C}$

• $\log \left|\mathrm{x}\right| + \mathrm{C}$

• $\log \left|\mathrm{x}\right| + 2{\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

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

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

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

• $\log \left|\log \left({\mathrm{x}}^2\right)\right| + \mathrm{C}$

• $2\log \left|\log \left({\mathrm{x}}^2\right)\right| + \mathrm{C}$

• $\frac{1}{4}\log \left|\log \left({\mathrm{x}}^2\right)\right| + \mathrm{C}$

${\int }_0^1\frac{1}{\left({\mathrm{x}}^2 + 16\right)\left({\mathrm{x}}^2 + 25\right)}\mathrm{dx}$ is equal to

• $\frac{1}{5}\left[\frac{1}{4}{\tan }^{-1}\left(\frac{1}{4}\right) - \frac{1}{5}{\tan }^{-1}\left(\frac{1}{5}\right)\right]$

• $\frac{1}{9}\left[\frac{1}{4}{\tan }^{-1}\left(\frac{1}{4}\right) - \frac{1}{5}{\tan }^{-1}\left(\frac{1}{5}\right)\right]$

• $\frac{1}{4}\left[\frac{1}{4}{\tan }^{-1}\left(\frac{1}{4}\right) - \frac{1}{5}{\tan }^{-1}\left(\frac{1}{5}\right)\right]$

• $\frac{1}{9}\left[\frac{1}{5}{\tan }^{-1}\left(\frac{1}{4}\right) - \frac{1}{4}{\tan }^{-1}\left(\frac{1}{5}\right)\right]$

${\int }_{- 1}^1\mathrm{x}\left(1 - \mathrm{x}\right)\left(1 +\hspace{0.17em}\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{1}{3}$

• $\frac{2}{3}$

• 1

• 0

The value of ${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}\frac{{\sin }^2\left(\mathrm{x}\right)}{1 + 7^{\mathrm{x}}}\mathrm{dx}$ is

• 7^x

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $2^{\mathrm{\pi }}$