$\int \frac{{\mathrm{x}}^2\mathrm{dx}}{{\left(\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right)}^2}$ is equal to

• $\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{c}$

• $\frac{x\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{c}$

• $\frac{\sin \left(\mathrm{x}\right) - \mathrm{xcos}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{c}$

• None of these

If f(x) = $\mathrm{Asin}\left(\frac{\mathrm{\pi x}}{2}\right) + \mathrm{B}, \mathrm{f}\text{'}\left(\frac{1}{2}\right) = \sqrt{2}$ and ${\int }_0^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \frac{2\mathrm{A}}{\mathrm{\pi }}$, then A and B are

• $\frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}$

• $\frac{2}{\mathrm{\pi }}, \frac{3}{\mathrm{\pi }}$

• $0, - 4\mathrm{\pi }$

• $\frac{4}{\mathrm{\pi }}, 0$

Let g(x) = ${\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$, where f is such that $\frac{1}{2} \le \mathrm{f}\left(\mathrm{x}\right) \le 1$ for t $\in$ [0, 1] and $0 \le \mathrm{f}\left(\mathrm{t}\right) \le \frac{1}{2}$ for t $\in$ [1, 2]. Then, g(2) satisfies the inequality

• $- \frac{3}{2} \le \mathrm{g}\left(2\right) < \frac{1}{2}$

• $0 \le \mathrm{g}\left(2\right) < 2$

• $\frac{1}{2} \le \mathrm{g}\left(2\right) < \frac{3}{2}$

• 2 < g(2) < 4

$\int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) + \sqrt{2}}$ is equal to

• $- \frac{1}{\sqrt{2}}\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right) + \mathrm{C}$

• $\frac{1}{\sqrt{2}}\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right) + \mathrm{C}$

• $\frac{1}{\sqrt{2}}\cot \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right) + \mathrm{C}$

• $\frac{1}{\sqrt{2}}\cot \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right) + \mathrm{C}$

$\int {\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{4}\right)\mathrm{dx}$ is equal to

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\cos \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• $\sqrt{2}{\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\cos \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• $\sqrt{2}{\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

The solution of the differential equation xdy - ydx = $\left(\sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right)\mathrm{dx}$ is

• $\mathrm{y} - \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{Cx}}^2$

• $\mathrm{y} + \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{Cx}}^2$

• $\mathrm{y} + \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} + {\mathrm{Cx}}^{2 }= 0$

• None of the above

707.

If ${\int }_0^{{\mathrm{t}}^2}\mathrm{x} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \frac{2}{5}{\mathrm{t}}^5,\mathrm{t} > 0, \mathrm{then} \mathrm{f}\left(\frac{4}{25}\right)$ is

• $\frac{2}{5}$

• $\frac{5}{2}$

• $- \frac{2}{5}$

• None of these

The value of interal I = $\int \frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\sqrt{1 + \sin \left(2\mathrm{x}\right)}}\mathrm{dx}$ is

• $\sqrt{1 + \cos \left(2\mathrm{x}\right)}$

• $\sqrt{\mathrm{x}}$

• x

• $\sqrt{1 + 2\mathrm{x}}$

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

• 0

• 1/2

• 1

• None of these

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

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

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

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

• None of the above