$\int {\cos }^3\left(\mathrm{x}\right) . {\mathrm{e}}^{\log \left(\sin \left(\mathrm{x}\right)\right)}\mathrm{dx}$ is equal to

• $- \frac{{\displaystyle {\sin }^4}\left(\mathrm{x}\right)}{4} + \mathrm{c}$

• $- \frac{{\displaystyle {\cos }^4\left(\mathrm{x}\right)}}{4} + \mathrm{c}$

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

• None of the above

The value of ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\cos \left(3\mathrm{x}\right) + 1}{2\cos \left(\mathrm{x}\right) - 1}\mathrm{dx}$ is

• 2

• 1

• $\frac{1}{2}$

• 0

The value of ${\int }_0^1{\tan }^{-1}\left(\frac{2\mathrm{x} - 1}{1 + \mathrm{x} - {\mathrm{x}}^2}\right)\mathrm{dx}$ is

• 1

• 0

• - 1

• None of the above

354.

By the application of Simpson's one - third rule for numerical integration, with two subintervals, the value of ${\int }_0^1\frac{\mathrm{dx}}{1 +\hspace{0.17em}\mathrm{x}}$ is

• $\frac{17}{36}$

• $\frac{17}{25}$

• $\frac{25}{36}$

• $\frac{17}{24}$

$\int \frac{{\mathrm{x}}^{\mathrm{e} - 1} + {\mathrm{e}}^{\mathrm{x} - 1}}{{\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}}\mathrm{dx}$ is equal to

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

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

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

• None of the above

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