If for every integer n, ${\int }_{\mathrm{n}}^{\mathrm{n} + 1}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = {\mathrm{n}}^2$, then the value of ${\int }_{- 2}^4\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is

• 16

• 14

• 19

• None of these

The value of integral ${\int }_0^{\mathrm{\pi }}\mathrm{xf}\left(\sin \left(\mathrm{x}\right)\right)d\mathrm{x}$ is

• 0

• $\mathrm{\pi }{\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{f}\left(\sin \left(\mathrm{x}\right)\right)d\mathrm{x}$

• $\frac{\mathrm{\pi }}{4}{\int }_0^{\mathrm{\pi }}\mathrm{f}\left(\sin \left(\mathrm{x}\right)\right)d\mathrm{x}$

• None of these

$\underset{\mathrm{x}\to \infty }{\lim }\frac{{\int }_0^{2\mathrm{x}}{\mathrm{xe}}^{\mathrm{x}}\mathrm{dx}}{{\mathrm{e}}^{4{\mathrm{x}}^2}}$ equals

• 0

• $\infty$

• 2

• $\frac{1}{2}$

The value of ${\int }_{- 2}^3\left|1 - {\mathrm{x}}^2\right|d\mathrm{x}$, is

• $\frac{1}{3}$

• 14/3

• 7/3

• 28/3

115.

$\int \frac{1}{\tan \left(\mathrm{x}\right) + \cot \left(\mathrm{x}\right) + \sec \left(\mathrm{x}\right) + \csc \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{1}{2}\left(\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right) + \mathrm{x} \right)$ + C

• $\frac{1}{2}\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) - \mathrm{x} \right)$ + C

• $\frac{1}{2}\left(\cos \left(\mathrm{x}\right) - \mathrm{x} + \sin \left(\mathrm{x}\right) \right)$ + C

• None of the above

If ${\int }_0^{\infty }\frac{\log \left(1 + {\mathrm{x}}^2\right)}{1 + {\mathrm{x}}^2}d\mathrm{x} = \mathrm{k}{\int }_0^1\frac{\log \left(1 + \mathrm{x}\right)}{1 + {\mathrm{x}}^2}\mathrm{dx}$, then k is equal to

• 4

• 8

• $\mathrm{\pi }$

• 2$\mathrm{\pi }$

${\int }_0^{\mathrm{x}}\left|\sin \left(\mathrm{t}\right)\right|\mathrm{dt}, \mathrm{where} \mathrm{x} \in \left[2\mathrm{n\pi }, \left(2\mathrm{n} + 1\right)\mathrm{\pi }\right], \mathrm{n} \in \mathrm{N}$, is equal to

• 4n - 1 - cos(x)

• 4n - sin(x)

• 4n - cos(x)

• 4n + 1 - cos(x)

Let I₁ = ${\int }_0^1\frac{{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}}{1 + \mathrm{x}}$ and I₂ = ${\int }_0^1\frac{{\mathrm{x}}^2\mathrm{dx}}{{\mathrm{e}}^{{\mathrm{x}}^3}\left(2 - {\mathrm{x}}^3\right)}$ Then, $\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}$ is equal to

• $\frac{1}{3\mathrm{e}}$

• 3e

• $\frac{\mathrm{e}}{3}$

• $\frac{3}{\mathrm{e}}$

${\int }_{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^5\frac{\sqrt{{\left(25 - {\mathrm{x}}^2\right)}^3}}{{\mathrm{x}}^4}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{5\mathrm{\pi }}{6}$

$\int \frac{\mathrm{dx}}{\cos \left(\mathrm{x}\right) + \sqrt{3}\sin \left(\mathrm{x}\right)}$ equals

• $\frac{1}{2}\log \left(\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{12}\right)\right) + \mathrm{C}$

• $\frac{1}{3}\log \left(\tan \left(\frac{\mathrm{x}}{2} - \frac{\mathrm{\pi }}{12}\right)\right) + \mathrm{C}$

• $\log \left(\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{6}\right)\right) + \mathrm{C}$

• $\frac{1}{2}\log \left(\tan \left(\frac{\mathrm{x}}{2} - \frac{\mathrm{\pi }}{6}\right)\right) + \mathrm{C}$