Let f(x) = ${\int }_0^{\mathrm{x}}\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$, where g is a non–zero even function. If f(x + 5) = g(x), then ${\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$ equals :

• ${\int }_{\mathrm{x} + 5}^5\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$

• $2{\int }_5^{\mathrm{x} + 5}\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$

• ${\int }_5^{\mathrm{x} +\hspace{0.17em}5}\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$

• $5{\int }_{\mathrm{x} + 5}^5\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$

If f : $\mathrm{R} \to \mathrm{R}$ is a differentiable function and f(2) = 6, then $\underset{\mathrm{x}\to 2}{\lim }{\int }_6^{\mathrm{f}\left(\mathrm{x}\right)}\frac{2\mathrm{t} \mathrm{dt}}{\left(\mathrm{x} - 2\right)}$ is

• 0

• 24f'(2)

• 12f'(2)

• 2f'(2)

The value of the integral ${\int }_0^1{\mathrm{xcot}}^{-1}\left(1 - {\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{x}}^4\right)\mathrm{dx}$ is

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

• $\frac{\mathrm{\pi }}{4} - {\log }_{\mathrm{e}}\left(2\right)$

• $\frac{\mathrm{\pi }}{4} - \frac{1}{2}{\log }_{\mathrm{e}}\left(2\right)$

• $\frac{\mathrm{\pi }}{2} - {\log }_{\mathrm{e}}\left(2\right)$

If $\int {\mathrm{e}}^{\sec \left(\mathrm{x}\right)}\left(\sec \left(\mathrm{x}\right)\tan \left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{x}\right) + \left(\sec \left(\mathrm{x}\right)\tan \left(\mathrm{x}\right) + {\sec }^2\left(\mathrm{x}\right)\right)\right)\mathrm{dx} = {\mathrm{e}}^{\sec \left(\mathrm{x}\right)}\mathrm{f}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$, then a possible choice of f(x) is :

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

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

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

• $\sec \left(\mathrm{x}\right) - \tan \left(\mathrm{x}\right) - \frac{1}{2}$

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

• $\frac{\mathrm{\pi } - 2}{4}$

• $\frac{\mathrm{\pi } - 2}{8}$

• $\frac{\mathrm{\pi } - 1}{4}$

• $\frac{\mathrm{\pi } - 1}{2}$

606.

The integral $\int {\sec }^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right){\csc }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $- 3 {\cot }^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) + \mathrm{C}$

• $- 3 {\tan }^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$

• $3 {\tan }^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$

• $- \frac{3}{4} {\tan }^{\raisebox{1ex}{$-4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$

If $\int \frac{\mathrm{dx}}{{\left({\mathrm{x}}^2 - 2\mathrm{x} + 10\right)}^2} = \mathrm{A}\left({\tan }^{-1}\left(\frac{\mathrm{x} - 1}{3}\right) + \frac{\mathrm{f}\left(\mathrm{x}\right)}{{\mathrm{x}}^2 - 2\mathrm{x} + 10}\right) + \mathrm{C}$ where C is a constant of integration, then

• $\mathrm{A} = \frac{1}{54}, \mathrm{f}\left(\mathrm{x}\right) = 3\left(\mathrm{x} - 1\right)$

• $\mathrm{A} = \frac{1}{54}, \mathrm{f}\left(\mathrm{x}\right) = 9{\left(\mathrm{x} - 1\right)}^2$

• $\mathrm{A} = \frac{1}{27}, \mathrm{f}\left(\mathrm{x}\right) = 9\left(\mathrm{x} - 1\right)$

• $\mathrm{A} = \frac{1}{81}, \mathrm{f}\left(\mathrm{x}\right) = 3\left(\mathrm{x} - 1\right)$

The value of ${\int }_0^{2\mathrm{\pi }}\left[\sin \left(2\mathrm{x}\right)\left(1 + \cos \left(3\mathrm{x}\right)\right)\right]\mathrm{dx}$, (where [t] denotes Greatest Integer Function)

• $- 2\mathrm{\pi }$

• $\mathrm{\pi }$

• $2\mathrm{\pi }$

• $- \mathrm{\pi }$

If $\int {\mathrm{x}}^5{\mathrm{e}}^{- {\mathrm{x}}^2}\mathrm{dx} = \mathrm{g}\left(\mathrm{x}\right){\mathrm{e}}^{- 2} + \mathrm{c}$, where c is a constant of interation then g(- 1) is equal to :

• - 1

• 1

• $\frac{1}{2}$

• $- \frac{5}{2}$

The integral ${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\sec }^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left({\csc }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right)\right)\mathrm{dx}$ =

• 3^5/3 - 3^1/3

• 3^7/6 - 3^5/6

• 3^5/6 - 3^1/3

• 3^4/3 - 3^1/3