The value of the integral ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{3 + \sin \left(2\mathrm{x}\right)}d\mathrm{x}$ is equal to

• ${\log }_{\mathrm{e}}\left(2\right)$

• ${\log }_{\mathrm{e}}\left(3\right)$

• $\frac{1}{4}{\log }_{\mathrm{e}}\left(2\right)$

• $\frac{1}{4}{\log }_{\mathrm{e}}\left(3\right)$

The value of the integral ${\int }_{- 2}^2\left(1 + 2\sin \left(\mathrm{x}\right)\right){\mathrm{e}}^{\left|\mathrm{x}\right|}d\mathrm{x}$ is equal to

• 0

• e² - 1

• 2(e² - 1)

• 1

The value of the integral ${\int }_1^5\left[\left|\mathrm{x} - 3\right| + \left|1 - \mathrm{x}\right|\right]d\mathrm{x}$ is equal to

• 4

• 8

• 12

• 16

354.

Let [x] denote the greatest integer less than or equal to x, then the value of the integral ${\int }_{- 1}^1\left(\left|\mathrm{x}\right| - 2\left[\mathrm{x}\right]\right)d\mathrm{x}$ is equal to

• 3

• 2

• - 2

• - 3

Prove that

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

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