Suppose $\mathrm{M} = {\int }_0^{\mathrm{\pi }/2}\frac{\cos \left(\mathrm{x}\right)}{\mathrm{x} + 2}d\mathrm{x}, \mathrm{N} = {\int }_0^{\mathrm{\pi }/4}\frac{\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}{{\left(\mathrm{x} + 1\right)}^2}d\mathrm{x}$. Then, the values of (M - N) equals

• $\frac{3}{\mathrm{\pi } + 2}$

• $\frac{2}{\mathrm{\pi } - 4}$

• $\frac{4}{\mathrm{\pi } - 2}$

• $\frac{2}{\mathrm{\pi } + 4}$

The value of the integral

${\int }_{- 1}^1\left\{\frac{{\mathrm{x}}^{2013}}{{\mathrm{e}}^{\left|\mathrm{x}\right|}\left({\mathrm{x}}^2 +\hspace{0.17em}\cos \left(\mathrm{x}\right)\right)} + \frac{1}{{\mathrm{e}}^{\left|\mathrm{x}\right|}}\right\}d\mathrm{x}$ is equal to

• 0

• $1 - {\mathrm{e}}^{- 1}$

• $2{\mathrm{e}}^{- 1}$

• $2\left(1 - {\mathrm{e}}^{- 1}\right)$

The value of I = ${\int }_0^{\mathrm{\pi }/4}\left({\tan }^{\mathrm{n} + 1}\left(\mathrm{x}\right)\right)d\mathrm{x} + \frac{1}{2}{\int }_0^{\mathrm{\pi }/2}\left({\tan }^{\mathrm{n} + 1}\left(\frac{\mathrm{x}}{2}\right)\right)d\mathrm{x}$ is

• $\frac{1}{\mathrm{n}}$

• $\frac{\mathrm{n} + 2}{2\mathrm{n} + 1}$

• $\frac{2\mathrm{n} - 1}{\mathrm{n}}$

• $\frac{2\mathrm{n} - 3}{3\mathrm{n} - 2}$

344.

The value of the integral

${\int }_1^2{\mathrm{e}}^{\mathrm{x}}\left({\log }_{\mathrm{e}}\left(\mathrm{x}\right) + \frac{\mathrm{x} + 1}{\mathrm{x}}\right)d\mathrm{x}$

• ${\mathrm{e}}^2\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

• ${\mathrm{e}}^2 - \mathrm{e}$

• ${\mathrm{e}}^2\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right) - \mathrm{e}$

• ${\mathrm{e}}^2 - \mathrm{e}\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

If [a] denote the greatest integer which is less than or equal to a. Then, the value of the integral ${\int }_{- \frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\left[\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)\right]d\mathrm{x}$ is

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $- \mathrm{\pi }$

• $- \frac{\mathrm{\pi }}{2}$

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

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{4}$

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.}$