If I = ${\int }_0^1\frac{\mathrm{dx}}{1 +\hspace{0.17em}{\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$, then

• ${\log }_{\mathrm{e}}\left(2\right) < 1 < \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• ${\log }_{\mathrm{e}}\left(2\right) > 1$

• $\mathrm{I} = \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\mathrm{I} = {\log }_{\mathrm{e}}\left(2\right)$

${\int }_0^{1000}{\mathrm{e}}^{\mathrm{x} - \left[\mathrm{x}\right]}d\mathrm{x}$ is equal to

• $\frac{{\mathrm{e}}^{1000} - 1}{\mathrm{e} - 1}$

• $\frac{{\mathrm{e}}^{1000} - 1}{1000}$

• $\frac{\mathrm{e} - 1}{1000}$

• 1000(e - 1)

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

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

• $\frac{1}{2}{\left({\sin }^{-1}\left(\mathrm{x}\right)\right)}^2 + \mathrm{c}$

• $\log \left(\sqrt{1 - {\mathrm{x}}^2}\right) + \mathrm{c}$

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

$\int \frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{x} + 1\right)}$ equals

• $\log \left|\frac{\mathrm{x} + 1}{\mathrm{x}}\right| + \mathrm{c}$

• $\log \left|\frac{\mathrm{x}}{\mathrm{x} + 1}\right| + \mathrm{c}$

• $\log \left|\frac{\mathrm{x} - 1}{\mathrm{x}}\right| + \mathrm{c}$

• $\log \left|\frac{\mathrm{x} - 1}{\mathrm{x} + 1}\right| + \mathrm{c}$

The value of integral ${\int }_{- 1}^1\frac{\left|\mathrm{x} + 2\right|}{\mathrm{x} + 2}d\mathrm{x}$ is

• 1

• 2

• 0

• - 1

$\int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}\right) + \sqrt{3}\cos \left(\mathrm{x}\right)}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{2} - \frac{\mathrm{\pi }}{6}\right)\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{4} - \frac{\mathrm{\pi }}{6}\right)\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{6}\right)\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{4} + \frac{\mathrm{\pi }}{3}\right)\right| + \mathrm{c}$

If f(x) = f(a - x), then ${\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is equal to

• ${\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $\frac{{\mathrm{a}}^2}{2}{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $\frac{\mathrm{a}}{2}{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $- \frac{\mathrm{a}}{2}{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

378.

The value of ${\int }_0^{\infty }\frac{d\mathrm{x}}{\left({\mathrm{x}}^2 + 4\right)\left({\mathrm{x}}^2 + 9\right)}$ is

• $\frac{\mathrm{\pi }}{60}$

• $\frac{\mathrm{\pi }}{20}$

• $\frac{\mathrm{\pi }}{40}$

• $\frac{\mathrm{\pi }}{80}$

If I₁ = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\sin }^2\left(\mathrm{x}\right)d\mathrm{x}$ and I₂ = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\cos }^2\left(\mathrm{x}\right)d\mathrm{x}$, then

• I₁ = I₂

• I₁ < I₂

• I₁ > I₂

• I₂ = I₁ + $\frac{\mathrm{\pi }}{4}$

The integrating factor of the differential equation $\mathrm{xlog}\left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = 2\log \left(\mathrm{x}\right)$ is given by

• e^x

• log(x)

• log(log(x))

• x