Let f(x) = x - [x], for every real x, where [x] is the greatest integer less than or equal to x. Then, ${\int }_{- 1}^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is :

• 1

• 2

• 3

• 0

If ${\int }_0^{{\mathrm{x}}^2}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt} = \mathrm{xcos}\left(\mathrm{\pi x}\right)$, then the value of f(4) is :

• 1

• $\frac{1}{4}$

• - 1

• $\frac{- 1}{4}$

If f(x) = $\frac{2 - \mathrm{xcos}\left(\mathrm{x}\right)}{2 + \mathrm{xcos}\left(\mathrm{x}\right)} \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)\hspace{0.17em}= {\log }_{\mathrm{e}}\left(\mathrm{x}\right), (\mathrm{x} > 0)$ then the value of integral ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{g}\left(\mathrm{f}\right(\mathrm{x}\left)\right)$dx is :

• log_e(3)

• log_e(1)

• log_e(2)

• log_ee

$\int \frac{\sin \left({\displaystyle \frac{5\mathrm{x}}{2}}\right)}{\sin \left({\displaystyle \frac{\mathrm{x}}{2}}\right)}\mathrm{dx}$ is equal to :

(where c is a constant of integration)

• $\mathrm{x} + 2\sin \left(\mathrm{x}\right) + 2\sin \left(2\mathrm{x}\right) + \mathrm{c}$

• $2\mathrm{x} + \sin \left(\mathrm{x}\right) + \sin \left(2\mathrm{x}\right) + \mathrm{c}$

• $2\mathrm{x} + \sin \left(2\mathrm{x}\right) + 2\sin \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{x} + \sin \left(2\mathrm{x}\right) + 2\sin \left(\mathrm{x}\right) + \mathrm{c}$

If $\int \frac{\mathrm{dx}}{{\mathrm{x}}^3{\left(1 + {\mathrm{x}}^6\right)}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}} = \mathrm{xf}\left(\mathrm{x}\right)(1 +\hspace{0.17em}{\mathrm{x}}^6{)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + \mathrm{C}$ where C is a constant of integration, then the function f(x) is equal to :

• - $\frac{1}{2{\mathrm{x}}^2}$

• $- \frac{1}{6{\mathrm{x}}^3}$

• $\frac{3}{{\mathrm{x}}^2}$

• $- \frac{1}{2{\mathrm{x}}^3}$

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 :

• $\frac{\mathrm{\pi } - 2}{4}$

• $\frac{\mathrm{\pi } - 2}{8}$

• $\frac{\mathrm{\pi } - 1}{4}$

• $\frac{\mathrm{\pi } - 1}{2}$