$\int \left(\mathrm{x} + 1\right){\left(\mathrm{x} + 2\right)}^7\left(\mathrm{x} + 3\right)\mathrm{dx}$ is equal to

• $\frac{{\left(\mathrm{x} + 2\right)}^{10}}{10} - \frac{{\displaystyle {\left(\mathrm{x} + 2\right)}^8}}{8} + \mathrm{c}$

• $\frac{{\left(\mathrm{x} + 1\right)}^{10}}{2} - \frac{{\displaystyle {\left(\mathrm{x} + 2\right)}^8}}{8} - \frac{{\left(\mathrm{x} + 3\right)}^2}{2}+ \mathrm{c}$

• $\frac{{\left(\mathrm{x} + 2\right)}^{10}}{10} + \mathrm{c}$

• $\frac{{\left(\mathrm{x} + 1\right)}^2}{2} + \frac{{\left(\mathrm{x} + 2\right)}^8}{8} + \frac{{\displaystyle {\left(\mathrm{x} + 3\right)}^2}}{2} + \mathrm{c}$

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

• $\frac{{\left(\mathrm{x} + 1\right)}^{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{7} - 2\frac{{\displaystyle {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{5} + 2\frac{{\displaystyle {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{3} + \mathrm{c}$

• $2\left[\frac{{\left(\mathrm{x} + 1\right)}^{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{7} - 2\frac{{\displaystyle {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{5} + 2\frac{{\displaystyle {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{3}\right] + \mathrm{c}$

• $\frac{{\left(\mathrm{x} + 1\right)}^{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{7} - 2\frac{{\displaystyle {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{5} + \mathrm{c}$

• ${\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + \mathrm{c}$

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

• $\log \left|\left(\mathrm{x} - {\mathrm{e}}^{- \mathrm{x}}\right)\right|$

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

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

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

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

• ${\left[\log \left(\mathrm{x} + \sqrt{1 + {\mathrm{x}}^2}\right)\right]}^2 + \mathrm{c}$

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

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

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

$\int \frac{\mathrm{dx}}{\sqrt{1 - {\mathrm{e}}^{2\mathrm{x}}}}$ is equal to

• $\log \left|{\mathrm{e}}^{- \mathrm{x}} + \sqrt{{\mathrm{e}}^{- 2\mathrm{x}} - 1}\right| + \mathrm{c}$

• $\log \left|{\mathrm{e}}^{\mathrm{x}} + \sqrt{{\mathrm{e}}^{2\mathrm{x}} - 1}\right| + \mathrm{c}$

• $- \log \left|{\mathrm{e}}^{- \mathrm{x}} + \sqrt{{\mathrm{e}}^{- 2\mathrm{x}} - 1}\right| + \mathrm{c}$

• $- \log \left|{\mathrm{e}}^{- 2\mathrm{x}} + \sqrt{{\mathrm{e}}^{- 2\mathrm{x}} - 1}\right| + \mathrm{c}$

466.

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

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

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

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

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

The integral ${\int }_0^1\frac{2{\sin }^{-1}\left({\displaystyle \frac{\mathrm{x}}{2}}\right)}{\mathrm{x}}\mathrm{dx}$ equals

• ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\frac{\mathrm{xdx}}{\tan \left(\mathrm{x}\right)}$

• ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\frac{2}{\tan \left(\mathrm{x}\right)}\mathrm{dx}$

• ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{2\mathrm{xdx}}{\tan \left(\mathrm{x}\right)}$

• ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\frac{2\mathrm{x}}{\sin \left(\mathrm{x}\right)}\mathrm{dx}$

If ${\int }_0^{\mathrm{a}}\mathrm{f}\left(2\mathrm{a} - \mathrm{x}\right)\mathrm{dx} = \mathrm{m} \mathrm{and} {\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{n}$, then ${\int }_0^{2\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• 2m + n

• m + 2n

• m - n

• m + n

${\int }_{- 100}^{100}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• ${\int }_{- 100}^{100}\mathrm{f}\left({\mathrm{x}}^2\right)\mathrm{dx}$

• ${\int }_{- 100}^{100}\mathrm{f}\left(- {\mathrm{x}}^2\right)\mathrm{dx}$

• ${\int }_{- 100}^{100}\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)\mathrm{dx}$

• ${\int }_{- 100}^{100}\mathrm{f}\left(- \mathrm{x}\right)\mathrm{dx}$

${\int }_{- 1}^1\left({\mathrm{e}}^{{\mathrm{x}}^3} + {\mathrm{e}}^{- {\mathrm{x}}^3}\right)\left({\mathrm{e}}^{\mathrm{x}} - {\mathrm{e}}^{- \mathrm{x}}\right)\mathrm{dx}$ is equal to

• $\frac{{\mathrm{e}}^2}{2} - 2\mathrm{e}$

• e² - 2e

• 2(e² - e)

• 0