The value of $\int \sqrt{1 + \sec \left(\mathrm{x}\right)}\mathrm{dx}$ is

• ${\sin }^{-1}\left(\sqrt{2}\sin \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{C}$

• $2{\sin }^{-1}\left(\sqrt{2}\sin \left(\mathrm{x}/2\right)\right) +\hspace{0.17em}\mathrm{C}$

• $2{\sin }^{-1}\left(\sqrt{2}\sin \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{C}$

• $2{\sin }^{-1}\left(\sqrt{2}\mathrm{x}/2\right) +\hspace{0.17em}\mathrm{C}$

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

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

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

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

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

673.

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

• $\frac{1}{32}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{32}$

The value of ${\int }_0^{\infty }\frac{\mathrm{x}}{\left(1 + \mathrm{x}\right)\left({\mathrm{x}}^2 + 1\right)}\mathrm{dx}$ is

• $2\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{32}$

$\int \frac{1}{\sqrt{8 + 2\mathrm{x} - {\mathrm{x}}^2}}\mathrm{dx}$ is equal to

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

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

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

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

$\int \left(\frac{4{\mathrm{e}}^{\mathrm{x}} - 25}{2{\mathrm{e}}^{\mathrm{x}} - 5}\right)\mathrm{dx} = \mathrm{Ax} + \mathrm{Blog}\left(2{\mathrm{e}}^{\mathrm{x}} - 5\right) + \mathrm{c}$, then

• A = 5 and B = 3

• A = 5 and B = - 3

• A = - 5 and B = 3

• A = - 5 and B = - 3

${\int }_{- \frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\log \left(\frac{2 - \sin \left(\mathrm{x}\right)}{2 + \sin \left(\mathrm{x}\right)}\right)\mathrm{dx}$ is equal to

• 1

• 3

• 2

• 0

$\int \left(\frac{\left({\mathrm{x}}^2 + 2\right){\mathrm{a}}^{\left(\mathrm{x} + {\tan }^{-1}\left(\mathrm{x}\right)\right)}}{{\mathrm{x}}^2 + 1}\right)\mathrm{dx}$ is equal to

• $\log \left(\mathrm{a}\right) . {\mathrm{a}}^{\mathrm{x} + {\tan }^{-1}\left(\mathrm{x}\right)} + \mathrm{c}$

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

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

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

If $\int \frac{\mathrm{f}\left(\mathrm{x}\right)}{\log \left(\sin \left(\mathrm{x}\right)\right)}\mathrm{dx} = \log \left[\log \left(\sin \left(\mathrm{x}\right)\right)\right] + \mathrm{c}$, then f(x) is equal to

• cot(x)

• tan(x)

• sec(x)

• csc(x)

${\int }_0^{\frac{\mathrm{\pi }}{2}}\left(\frac{\sqrt[\mathrm{n}]{\sec \left(\mathrm{x}\right)}}{\sqrt[\mathrm{n}]{\sec \left(\mathrm{x}\right)} +\sqrt[\mathrm{n}]{\csc \left(\mathrm{x}\right)}}\right)\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$