$\int \left(\sqrt[3]{\mathrm{x}}\right)\left(\sqrt[5]{1 + \sqrt[3]{{\mathrm{x}}^4}}\right)\mathrm{dx}$ is equal to

• ${\left(1 + {\mathrm{x}}^{\frac{3}{4}}\right)}^{\frac{6}{5}} + \mathrm{c}$

• ${\left(1 + {\mathrm{x}}^{\frac{4}{3}}\right)}^{\frac{6}{5}} + \mathrm{c}$

• $\frac{5}{8}{\left(1 + {\mathrm{x}}^{\frac{4}{3}}\right)}^{\frac{6}{5}} + \mathrm{c}$

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

If u = - $\mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right)\sin \left(\mathrm{\theta }\right) + \mathrm{f}\text{'}\left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right)$ and v = $\mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right) + \mathrm{f}\text{'}\left(\mathrm{\theta }\right)\sin \left(\mathrm{\theta }\right)$, then $\int {\left[{\left(\frac{\mathrm{du}}{\mathrm{d\theta }}\right)}^2 + {\left(\frac{\mathrm{dv}}{\mathrm{d\theta }}\right)}^2\right]}^{\frac{1}{2}}\mathrm{d\theta }$ is equal to

• $\mathrm{f}\left(\mathrm{\theta }\right) - \mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right) + \mathrm{c}$

• $\mathrm{f}\left(\mathrm{\theta }\right) + \mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right) + \mathrm{c}$

• $\mathrm{f}\text{'}\left(\mathrm{\theta }\right) + \mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right) + \mathrm{c}$

• $\mathrm{f}\text{'}\left(\mathrm{\theta }\right) - \mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right) + \mathrm{c}$

$\int \frac{{\mathrm{e}}^{6{\log }_{\mathrm{e}}\left(\mathrm{x}\right)} - {\mathrm{e}}^{5{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}}{{\mathrm{e}}^{4{\log }_{\mathrm{e}}\left(\mathrm{x}\right)} - {\mathrm{e}}^{3{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}}\mathrm{dx}$ is equal to

• $\frac{{\mathrm{x}}^3}{3} + \mathrm{c}$

• $\frac{{\mathrm{x}}^2}{2} + \mathrm{c}$

• $\frac{{\mathrm{x}}^2}{3} + \mathrm{c}$

• $\frac{- {\mathrm{x}}^3}{3} + \mathrm{c}$

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

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

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

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

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

$\int \frac{{\mathrm{x}}^4 - 1}{{\mathrm{x}}^2{\left({\mathrm{x}}^4 + {\mathrm{x}}^2 + 1\right)}^{{\displaystyle \frac{1}{2}}}}\mathrm{dx}$ is equal to

• $\sqrt{\frac{{\mathrm{x}}^4 + {\mathrm{x}}^2 + 1}{\mathrm{x}}} + \mathrm{c}$

• $\frac{{\mathrm{x}}^2}{\sqrt{{\mathrm{x}}^4 + {\mathrm{x}}^2 + 1}} + \mathrm{c}$

• $\mathrm{x}{\left({\mathrm{x}}^4 + {\mathrm{x}}^2 + 1\right)}^{\frac{3}{2}} + \mathrm{c}$

• $\frac{\sqrt{{\mathrm{x}}^4 + {\mathrm{x}}^2 + 1}}{\mathrm{x}} + \mathrm{c}$

456.

$\int \frac{\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)}{1 + 2\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}\mathrm{dx}$ 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${\log }_{\mathrm{e}}\left(10\mathrm{e}\right)$

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

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

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

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