If $\int \sin \left\{2{\tan }^{-1}\left(\sqrt{\frac{1 - \mathrm{x}}{1 + \mathrm{x}}}\right)\right\}\mathrm{dx} = {\mathrm{Asin}}^{-1}\left(\mathrm{x}\right) + \mathrm{Bx}\sqrt{1 - {\mathrm{x}}^2} + \mathrm{C}$, then A + B is equal to

• 10

• $\frac{1}{2}$

• 1

• $- \frac{1}{2}$

${\int }_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{e}$}\right.}^1\left|\log \left(\mathrm{x}\right)\right|\mathrm{dx}$ is equal to

• $\frac{1}{\mathrm{e}}$

• e

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

• None of the above

$\underset{\mathrm{x}\to 0}{\lim }\frac{{\int }_0^{{\mathrm{x}}^2}\sin \left(\sqrt{\mathrm{t}}\right)\mathrm{dt}}{{\mathrm{x}}^3}$ is equal to

• $\frac{2}{3}$

• $\frac{1}{3}$

• 0

• $\infty$

By trapezoidal rule, the approximate value of the integral ${\int }_0^6\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}$ is

• 1.3128

• 1.4108

• 1.4218

• None of these

885.

The value of the integral $\mathrm{I} = \int \left(\sqrt{\tan \left(\mathrm{x}\right)} + \sqrt{\cot \left(\mathrm{x}\right)}\right)\mathrm{dx}, \mathrm{where} \mathrm{x} \in \left(0, \frac{\mathrm{\pi }}{2}\right)$, is

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

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

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

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

The value of ${\int }_0^{\sqrt{\ln \left(\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}}\cos \left({\mathrm{e}}^{\mathrm{x}}\right)2{\mathrm{xe}}^{{\mathrm{x}}^2}\mathrm{dx}$ is

• 1

• 1 + sin(1)

• 1 - sin(1)

• (sin(1) - 1

If f(x) = ${\int }_2^{\mathrm{x}}\frac{\mathrm{dt}}{\sqrt{1 + {\mathrm{t}}^4}}$ and g is the inverse of f. Then, the value of g'(0) is

• 1

• 17

• $\sqrt{17}$

• None of the above

${\int }_0^{100}{\mathrm{e}}^{\mathrm{x} - \left[\mathrm{x}\right]}\mathrm{dx}$ is equal to

• 50(e - 1)

• 75(e - 1)

• 90(e - 1)

• 100(e - 1)

By Simpson's $\frac{1}{3}$rd rule, the approximate value of the integral ${\int }_1^2{\mathrm{e}}^{\raisebox{1ex}{$- \mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{dx}$ using four intervals, is

• 0.377

• 0.487

• 0.477

• 0.387

For n = 4, using trapezoidal rule, the value of ${\int }_0^2\frac{\mathrm{dx}}{1 + \mathrm{x}}$ will be

• 1.116625

• 1.1176

• 1.1180

• None of these