The value of ${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathrm{e}}^{\mathrm{x}}\left(\log \left(\sin \left(\mathrm{x}\right)\right) + \cot \left(\mathrm{x}\right)\right)\mathrm{dx}$ is

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\log \left(2\right)$

• $- {\mathrm{e}}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\log \left(2\right)$

• $\frac{1}{2}{\mathrm{e}}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\log \left(2\right)$

• $- \frac{1}{2}{\mathrm{e}}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\log \left(2\right)$

Considering four sub-intervals, the value of ${\int }_0^1\frac{1}{1 +\hspace{0.17em}\mathrm{x}}\mathrm{dx}$ by Trapezoidal rule, is

• 0.6870

• 0.6677

• 0.6977

• 0.5970

By Simpson's rule, the value of ${\int }_1^2\frac{\mathrm{dx}}{\mathrm{x}}$ dividing the interval (1, 2) into four equal parts, is

• 0.6932

• 0.6753

• 0.6692

• 7.1324

The value of $\int \mathrm{xsin}\left(\mathrm{x}\right){\sec }^3\left(\mathrm{x}\right)\mathrm{dx}$ is

• $\frac{1}{2}\left[{\sec }^2\left(\mathrm{x}\right) - \tan \left(\mathrm{x}\right)\right] + \mathrm{c}$

• $\frac{1}{2}\left[{\mathrm{xsec}}^2\left(\mathrm{x}\right) - \tan \left(\mathrm{x}\right)\right] + \mathrm{c}$

• $\frac{1}{2}\left[{\mathrm{xsec}}^2\left(\mathrm{x}\right) + \tan \left(\mathrm{x}\right)\right] + \mathrm{c}$

• $\frac{1}{2}\left[{\sec }^2\left(\mathrm{x}\right) + \tan \left(\mathrm{x}\right)\right] + \mathrm{c}$

The value of ${\int }_0^{\mathrm{x}}{\mathrm{xsin}}^3\left(\mathrm{x}\right)\mathrm{dx}$ is

• $\frac{4\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

• 0

• None of these

Which of the following is true ?

• ${\int }_0^1{\mathrm{e}}^{\mathrm{x}}\mathrm{dx} = \mathrm{e}$

• ${\int }_0^1{2}^{\mathrm{x}}\mathrm{dx} = \log \left(2\right)$

• ${\int }_0^1\sqrt{\mathrm{x}}\mathrm{dx} = \frac{2}{3}$

• ${\int }_0^1\mathrm{xdx} = \frac{1}{3}$

647.

$\int \left[\sin \left(\log \left(\mathrm{x}\right)\right) + \cos \left(\log \left(\mathrm{x}\right)\right)\right]\mathrm{dx}$ is equal to

• $\mathrm{xcos}\left(\log \left(\mathrm{x}\right)\right) + \mathrm{c}$

• $\cos \left(\log \left(\mathrm{x}\right)\right) + \mathrm{c}$

• $\mathrm{xsin}\left(\log \left(\mathrm{x}\right)\right) + \mathrm{c}$

• $\sin \left(\log \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{c}$

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

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

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

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

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

${\int }_5^{10}\frac{1}{\left(\mathrm{x} - 1\right)\left(\mathrm{x} - 2\right)}\mathrm{dx}$

• $\log \left(\frac{27}{32}\right)$

• $\log \left(\frac{32}{27}\right)$

• $\log \left(\frac{8}{9}\right)$

• $\log \left(\frac{3}{4}\right)$

$\int \mathrm{xlog}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

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

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

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

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