The value of ${\int }_{- 2}^2\left(\mathrm{xcos}\left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right) + 1\right)d\mathrm{x}$

• 2

• 0

• - 2

• 4

${\int }_{\mathrm{\pi }}^{16\mathrm{\pi }}\left|\sin \left(\mathrm{x}\right)\right|d\mathrm{x}$ is equal to

• 0

• 32

• 30

• 28

73.

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

• $2\sin \left(\mathrm{x}\right) + \log \left|\sec \left(\mathrm{x}\right) +\hspace{0.17em}\tan \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $2\sin \left(\mathrm{x}\right) - \log \left|\sec \left(\mathrm{x}\right) -\hspace{0.17em}\tan \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $2\sin \left(\mathrm{x}\right) - \log \left|\sec \left(\mathrm{x}\right) +\hspace{0.17em}\tan \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $2\sin \left(\mathrm{x}\right) + \log \left|\sec \left(\mathrm{x}\right) -\hspace{0.17em}\tan \left(\mathrm{x}\right)\right| + \mathrm{C}$

$\int \frac{{\sin }^8\left(\mathrm{x}\right) - {\cos }^8\left(\mathrm{x}\right)}{1 - 2{\sin }^2\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$

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

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

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

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

The value of ${\int }_0^{\mathrm{\pi }}{\sin }^{50}\left(\mathrm{x}\right){\cos }^{49}\left(\mathrm{x}\right)d\mathrm{x}$ is

• 0

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• 1

$\int 2^{\mathrm{x}}\left(\mathrm{f}\text{'}\left(\mathrm{x}\right) + \mathrm{f}\left(\mathrm{x}\right)\log \left(2\right)\right)\mathrm{dx}$ is

• 2^xf'(x) + C

• 2^xf(x) + C

• 2^x(log(2))f(x) + C

• log(2)f(x) + C

Evaluate the following integral

${\int }_{- 1}^2\left|\mathrm{xsin}\left(\mathrm{\pi x}\right)\right|d\mathrm{x}$

$\int \frac{\log \left(\sqrt{\mathrm{x}}\right)}{3\mathrm{x}}\mathrm{dx}$ is equal to

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

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

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

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

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

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

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

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

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

The value of the integral $\int \frac{\mathrm{dx}}{{\left({\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{- \mathrm{x}}\right)}^2}$

• $\frac{1}{2}\left({\mathrm{e}}^{2\mathrm{x}} +\hspace{0.17em}1\right) + \mathrm{c}$

• $\frac{1}{2}\left({\mathrm{e}}^{- 2\mathrm{x}} +\hspace{0.17em}1\right) + \mathrm{c}$

• $- \frac{1}{2}{\left({\mathrm{e}}^{2\mathrm{x}} +\hspace{0.17em}1\right)}^{- 1} + \mathrm{c}$

• $\frac{1}{4}\left({\mathrm{e}}^{2\mathrm{x}} -\hspace{0.17em}1\right) + \mathrm{c}$