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

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

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

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

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

$\int \frac{1}{\mathrm{x}}{\log }_{\mathrm{ex}}\left(\mathrm{e}\right)\mathrm{dx}$ is equal to

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

• ${\log }_{\mathrm{e}}\left({\log }_{\mathrm{e}}\left(\mathrm{ex}\right) - 1\right)$ + c

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

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

The value of ${\int }_1^{\mathrm{e}}{10}^{{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}d\mathrm{x}$ is equal to

• 10${\log }_{\mathrm{e}}\left(10\mathrm{e}\right)$

• $\frac{10\mathrm{e} - 1}{{\log }_{\mathrm{e}}\left(10\mathrm{e}\right)}$

• $\frac{10\mathrm{e}}{{\log }_{\mathrm{e}}\left(10\mathrm{e}\right)}$

• $\left(10\mathrm{e}\right){\log }_{\mathrm{e}}\left(10\mathrm{e}\right)$

174.

The value of ${\int }_{- 2}^4\left|\mathrm{x} + 1\right|\mathrm{dx}$ is equal to

• 12

• 14

• 13

• 16

If $\int \frac{\mathrm{x} + 2}{2{\mathrm{x}}^2 + 6\mathrm{x} + 5}\mathrm{dx}$ = P $\int \frac{4\mathrm{x} + 6}{2{\mathrm{x}}^2 + 6\mathrm{x} + 5}\mathrm{dx}$ + $\frac{1}{2}\int \frac{{\displaystyle \mathrm{dx}}}{{\displaystyle 2{\mathrm{x}}^2 + 6\mathrm{x} + 5}}$, then the values of P is

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{1}{4}$

• 2

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

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

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

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

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