$\int \frac{\mathrm{x} + 2}{\left({\mathrm{x}}^2 +\hspace{0.17em}3\mathrm{x} +\hspace{0.17em}3\right)\sqrt{\mathrm{x} + 1}}\mathrm{dx}$ is equa to

• $\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{x} + 1}\right) + \mathrm{C}$

• $\frac{2}{\sqrt{3}}{\tan }^{-1}\left[\frac{\mathrm{x}}{\sqrt{3\left(\mathrm{x} + 1\right)}}\right] + \mathrm{C}$

• $\frac{2}{\sqrt{3}}{\tan }^{-1}\left[\frac{\mathrm{x}}{{\left(\mathrm{x} + 1\right)}^2}\right] + \mathrm{C}$

• None of these

$\int {\sin }^{-1}\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a} + \mathrm{x}}}\right)\mathrm{dx}$ is equal to

• $\left[{\tan }^{-1}\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}}} + \sqrt{\frac{\mathrm{x}}{\mathrm{a}}}\right)\right] + \mathrm{C}$

• $\mathrm{a}\left[{\tan }^{-1}\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}}} - \sqrt{\frac{\mathrm{x}}{\mathrm{a}}}\right)\right] + \mathrm{C}$

• $\mathrm{a}\left[{\tan }^{-1}\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}}} . \frac{\left(\mathrm{a} +\hspace{0.17em}\mathrm{x}\right)}{\mathrm{a}}\right)\right] + \mathrm{C}$

• $\mathrm{a}\left[{\tan }^{-1}\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}}} . \frac{\left(\mathrm{a} + \mathrm{x}\right)}{\mathrm{a}} - \sqrt{\frac{\mathrm{x}}{\mathrm{a}}}\right)\right] + \mathrm{C}$

$\underset{\mathrm{n}\to \infty }{\lim }\frac{1}{\mathrm{n}}\left(\frac{1}{\mathrm{n} + 1} + \frac{2}{\mathrm{n} + 2} + ... + \frac{3\mathrm{n}}{4\mathrm{n}}\right)$ is equal to

• log(4)

• - log(4)

• 1 - log(4)

• None of these

The value of the integral ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{{\sin }^2\left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) +\hspace{0.17em}\cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\sqrt{2}\left(\log \left(\sqrt{2}\right)\right)$

• $\sqrt{2}\left(\sqrt{2} + 1\right)$

• $\log \left(\sqrt{2} + 1\right)$

• None of the above

$\int \frac{\mathrm{dx}}{9 + 16{\sin }^2\left(\mathrm{x}\right)}$ is equal to

• $\frac{1}{3}{\tan }^{-1}\left(\frac{3\tan \left(\mathrm{x}\right)}{5}\right) + \mathrm{c}$

• $\frac{1}{5}{\tan }^{-1}\left(\frac{\tan \left(\mathrm{x}\right)}{15}\right) + \mathrm{c}$

• $\frac{1}{15}{\tan }^{-1}\left(\frac{\tan \left(\mathrm{x}\right)}{5}\right) + \mathrm{c}$

• $\frac{1}{15}{\tan }^{-1}\left(\frac{5\tan \left(\mathrm{x}\right)}{3}\right) + \mathrm{c}$

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

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

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

• $0, - 4\mathrm{\pi }$

• $\frac{4}{\mathrm{\pi }}, 0$

418.

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}$