If I₁ = $\int {\sin }^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ and I₂ = $\int {\sin }^{-1}\left(\sqrt{1 - {\mathrm{x}}^2}\right)\mathrm{dx}$, then

• I₁ = I₂

• ${\mathrm{I}}_2 = \frac{\mathrm{\pi }}{2}{\mathrm{I}}_1$

• ${\mathrm{I}}_1 + {\mathrm{I}}_{2 }= \frac{\mathrm{\pi }}{2}\mathrm{x}$

• ${\mathrm{I}}_1 + {\mathrm{I}}_{2 }= \frac{\mathrm{\pi }}{2}$

$\int \frac{\left(\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)\right)}{\sqrt{\sin \left(2\mathrm{\theta }\right)}}\mathrm{d\theta }$ is equal to

• $\log \left|\cos \left(\mathrm{\theta }\right) - \sin \left(\mathrm{\theta }\right) + \sqrt{\sin \left(2\mathrm{\theta }\right)}\right| + \mathrm{c}$

• $\log \left|\sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right) + \sqrt{\sin \left(2\mathrm{\theta }\right)}\right| + \mathrm{c}$

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

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

${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{\mathrm{dx}}{1 + \sqrt{\tan \left(\mathrm{x}\right)}}$ is equal to

• $\frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

If f is a continuous function, then

• ${\int }_{- 2}^2\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_0^2\left[\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}(- \mathrm{x})\right]\mathrm{dx}$

• ${\int }_{- 3}^{5}2\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_{- 6}^{10}\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

• ${\int }_{- 3}^5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_{- 4}^4\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

• ${\int }_{- 3}^5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_{- 2}^6\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

$\int \frac{1 + \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• $\log \left(1 + \cos \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{c}$

• $\cot \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

• $\log \left(\mathrm{x} + \sin \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{c}$

$\int {\cos }^3\left(\mathrm{x}\right) . {\mathrm{e}}^{\log \left(\sin \left(\mathrm{x}\right)\right)}\mathrm{dx}$ is equal to

• $- \frac{{\displaystyle {\sin }^4}\left(\mathrm{x}\right)}{4} + \mathrm{c}$

• $- \frac{{\displaystyle {\cos }^4\left(\mathrm{x}\right)}}{4} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}{4} +\hspace{0.17em}\mathrm{c}$

• None of the above

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

• 2

• 1

• $\frac{1}{2}$

• 0

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

• 1

• 0

• - 1

• None of the above

639.

By the application of Simpson's one - third rule for numerical integration, with two subintervals, the value of ${\int }_0^1\frac{\mathrm{dx}}{1 +\hspace{0.17em}\mathrm{x}}$ is

• $\frac{17}{36}$

• $\frac{17}{25}$

• $\frac{25}{36}$

• $\frac{17}{24}$

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

• $\log \left({\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}\right) +\hspace{0.17em}\mathrm{c}$

• $\mathrm{elog}\left({\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}\right) +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{\mathrm{e}}\log \left({\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}\right) +\hspace{0.17em}\mathrm{c}$

• None of the above