Mathematics : Integrals

$\int 2^{\mathrm{x}}\left(\mathrm{f}\text{'}\left(\mathrm{x}\right) + \mathrm{f}\left(\mathrm{x}\right)\log \left(2\right)\right)\mathrm{dx}$ is

• 2^xf'(x) + C

• 2^xf(x) + C

• 2^x(log(2))f(x) + C

• log(2)f(x) + C

Evaluate the following integral

${\int }_{- 1}^2\left|\mathrm{xsin}\left(\mathrm{\pi x}\right)\right|d\mathrm{x}$

$\int \frac{\log \left(\sqrt{\mathrm{x}}\right)}{3\mathrm{x}}\mathrm{dx}$ is equal to

• $\frac{1}{3}{\left(\log \left(\sqrt{\mathrm{x}}\right)\right)}^2 + \mathrm{c}$

• $\frac{2}{3}{\left(\log \left(\sqrt{\mathrm{x}}\right)\right)}^2 + \mathrm{c}$

• $\frac{2}{3}{\left(\log \left(\mathrm{x}\right)\right)}^2 + \mathrm{c}$

• $\frac{1}{3}{\left(\log \left(\mathrm{x}\right)\right)}^2 + \mathrm{c}$

$\int {\mathrm{e}}^{\mathrm{x}}\left(\frac{2}{\mathrm{x}} - \frac{2}{{\mathrm{x}}^2}\right)\mathrm{dx}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{2{\mathrm{x}}^2} + \mathrm{c}$

• $\frac{2{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} + \mathrm{c}$

• $\frac{2{\mathrm{e}}^{\mathrm{x}}}{{\mathrm{x}}^2} + \mathrm{c}$

The value of the integral $\int \frac{\mathrm{dx}}{{\left({\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{- \mathrm{x}}\right)}^2}$

• $\frac{1}{2}\left({\mathrm{e}}^{2\mathrm{x}} +\hspace{0.17em}1\right) + \mathrm{c}$

• $\frac{1}{2}\left({\mathrm{e}}^{- 2\mathrm{x}} +\hspace{0.17em}1\right) + \mathrm{c}$

• $- \frac{1}{2}{\left({\mathrm{e}}^{2\mathrm{x}} +\hspace{0.17em}1\right)}^{- 1} + \mathrm{c}$

• $\frac{1}{4}\left({\mathrm{e}}^{2\mathrm{x}} -\hspace{0.17em}1\right) + \mathrm{c}$

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

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

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

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

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

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