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

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

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

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

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

82.

The value of integral ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sin }^5\left(\mathrm{x}\right)d\mathrm{x}$ is

• $\frac{4}{15}$

• $\frac{8}{5}$

• $\frac{8}{15}$

• $\frac{4}{5}$

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

• $\frac{1}{2}\left[{\mathrm{f}}^2\left(\mathrm{b}\right) - {\mathrm{f}}^2\left(\mathrm{a}\right)\right]$

• $\frac{1}{2}\left[{\mathrm{g}}^2\left(\mathrm{b}\right) - {\mathrm{g}}^2\left(\mathrm{a}\right)\right]$

• f(b) - f(a)

• $\frac{1}{2}\left[\mathrm{f}\left({\mathrm{b}}^2\right) - \mathrm{f}\left({\mathrm{a}}^2\right)\right]$

If I₁ = ${\int }_0^{3\mathrm{\pi }}\mathrm{f}\left({\cos }^2\left(\mathrm{x}\right)\right)d\mathrm{x}$

and I₂$= {\int }_0^{\mathrm{\pi }}\mathrm{f}\left({\cos }^2\left(\mathrm{x}\right)\right)d\mathrm{x}$, then

• I₁ = I₂

• 3I₁ = I₂

• I₁ = 3I₂

• I₁ = 5I₂

The value of I = ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left|\sin \left(\mathrm{x}\right)\right|d\mathrm{x}$ is

• 0

• 2

• - 2

• - 2 < 1 < 2

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