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

14.

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

Area bounded by the lines y = x, x = - 1, x = 2 and x - axis is

• 5/2 sq unit

• 3/2 sq unit

• 1/2 sq unit

• None of the above

The symmetric equation of lines 3x + 2y + z - 5 = 0 and x + y - 2z - 3 = 0, is

• $\frac{\mathrm{x} - 1}{5} = \frac{\mathrm{y} - 4}{7} = \frac{\mathrm{z} - 0}{1}$

• $\frac{\mathrm{x} + 1}{5} = \frac{\mathrm{y} + 4}{7} = \frac{\mathrm{z} - 0}{1}$

• $\frac{\mathrm{x} + 1}{- 5} = \frac{\mathrm{y} - 4}{7} = \frac{\mathrm{z} - 0}{1}$

• $\frac{\mathrm{x} - 1}{- 5} = \frac{\mathrm{y} - 4}{7} = \frac{\mathrm{z} - 0}{1}$

   

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} - \mathrm{y} + 3}{2\left(\mathrm{x} - \mathrm{y}\right) + 5}$ is

• $2\left(\mathrm{x} - \mathrm{y}\right) + \log \left(\mathrm{x} - \mathrm{y}\right) = \mathrm{x} \hspace{0.17em}+ \mathrm{c}$

• $2\left(\mathrm{x} - \mathrm{y}\right) - \log \left(\mathrm{x} - \mathrm{y} + 2\right) = \mathrm{x} \hspace{0.17em}+ \mathrm{c}$

   

• $2\left(\mathrm{x} - \mathrm{y}\right) + \log \left(\mathrm{x} - \mathrm{y} + 2\right) = \mathrm{x} \hspace{0.17em}+ \mathrm{c}$

• None of the above

$\overrightarrow{\mathrm{a}} \times \left[\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)\right]$ is equal to

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{a}}\right) . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}\right)$

• $\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}\right) - \overrightarrow{\mathrm{b}}\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$

• $\left[\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)\right]\overrightarrow{\mathrm{a}}$

• $\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{a}}\right) \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}}\right)$

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