The value of integral ${\int }_{\frac{\mathrm{\pi }}{6}}^{\frac{\mathrm{\pi }}{3}}\frac{\left(\sin \left(\mathrm{x}\right) - \mathrm{xcos}\left(\mathrm{x}\right)\right)}{\mathrm{x}\left(\mathrm{x} + \sin \left(\mathrm{x}\right)\right)}d\mathrm{x}$

• ${\log }_{\mathrm{e}}\left(\frac{2\left(\mathrm{\pi } + 3\right)}{\left(2\mathrm{\pi } + 3\sqrt{3}\right)}\right)$

• ${\log }_e\left(\frac{\mathrm{\pi } + 3}{2\left(2\mathrm{\pi } + 3\sqrt{3}\right)}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{2\mathrm{\pi } + 3\sqrt{3}}{2\left(\mathrm{\pi } + 3\right)}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{2\left(2\mathrm{\pi } + 3\sqrt{3}\right)}{\mathrm{\pi } +\hspace{0.17em}3}\right)$

If F(x) = ${\int }_0^{\mathrm{x}}\frac{\cos \left(\mathrm{t}\right)}{\left(1 +\hspace{0.17em}{\mathrm{t}}^2\right)}d\mathrm{t}, 0 \le \mathrm{x} \le 2\mathrm{\pi }$. Then,

• F is decreasing in $\left(\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2}\right)$ and decreasing in $\left(0, \frac{\mathrm{\pi }}{2}\right) \mathrm{and} \left(\frac{3\mathrm{\pi }}{2}, 2\mathrm{\pi }\right)$

• F is increasing in (0, $\mathrm{\pi }$) and decreasing in ( $\mathrm{\pi }, 2\mathrm{\pi }$).

• Fis increasing in ($\mathrm{\pi }, 2\mathrm{\pi }$) and decreasing in ( $0, \mathrm{\pi }$).

• Fis increasing in (0, $\frac{\mathrm{\pi }}{2}$) and $\left(\frac{3\mathrm{\pi }}{2}, 2\mathrm{\pi }\right)$ and decreasing in ( $\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2}$).

The value of the integral ${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(\frac{1 +\hspace{0.17em}\sin \left(2\mathrm{x}\right) + \cos \left(2\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}\right)d\mathrm{x}$ is equal to

• 16

• 8

• 4

• 1

The value of the integral ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{1}{1 +\hspace{0.17em}{\left(\tan \left(\mathrm{x}\right)\right)}^{101}}d\mathrm{x}$ is equal to

• 1

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{4}$

65.

The integrating factor of the differential equation

$3{\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = 2{\log }_{\mathrm{e}}\left(\mathrm{x}\right)$ is given by

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

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

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

• ${\left({\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

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

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