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

602.

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

The value of ${\int }_0^5\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}$ by chosing six sub-intervals and by using Simpson's rule will be

• 1.3562

• 1.3662

• 1.3456

• 1.2662

If $\int \frac{\cos \left(4\mathrm{x}\right) + 1}{\cot \left(\mathrm{x}\right) - \tan \left(\mathrm{x}\right)}\mathrm{dx} = \mathrm{Acos}\left(4\mathrm{x}\right) + \mathrm{B}$, then the value of A is

• $\frac{1}{2}$

• $\frac{1}{8}$

• $- \frac{1}{8}$

• $\frac{1}{4}$

If f(x) = $\mathrm{f}\left(\mathrm{x}\right) = \sqrt{\mathrm{x}}, \mathrm{g}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}} - 1$ and $\int \mathrm{fog}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{Afog}\left(\mathrm{x}\right) + {\mathrm{Btan}}^{-1}\left(\mathrm{fog}\left[\mathrm{x}\right]\right) + \mathrm{C}$, then the value of A + B is

• 1

• 2

• 3

• None of these

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

• 0

• 1

• - 1

• None of these

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(2\mathrm{x}\right){\tan }^{-1}\left(\sin \left(\mathrm{x}\right)\right)\mathrm{dx}$ =

• 1

• 0

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} - 1$