Mathematics : Integrals

Let f(x) = x² - 2. If ${\int }_3^6\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = 3\mathrm{f}\left(\mathrm{c}\right)$ for some c $\in$ (3, 6),  then the value of c is equal to

• $\sqrt{12}$

• $\sqrt{21}$

• $\sqrt{19}$

• $\sqrt{17}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\sin \left(2\mathrm{x}\right)}{1 + 2{\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{1}{2}\log \left(2\right)$

• $\log \left(2\right)$

• $\frac{1}{2}\log \left(3\right)$

• $\log \left(3\right)$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\mathrm{dx}}{1 +\hspace{0.17em}{\tan }^3\left(\mathrm{x}\right)}$ is equal to

• 1

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

If f(x) = ${\int }_1^{\mathrm{x}}{\sin }^2\left(\frac{\mathrm{t}}{2}\right)\mathrm{dt}$, then the value of $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{\pi } +\hspace{0.17em}\mathrm{x}\right) - \mathrm{f}\left(\mathrm{\pi }\right)}{\mathrm{x}}$ is equal to

• $\frac{1}{4}$

• $\frac{1}{2}$

• $\frac{3}{4}$

• 1

$\int \frac{1}{{\mathrm{x}}^2{\left({\mathrm{x}}^4 + 1\right)}^{{\displaystyle \frac{3}{4}}}}\mathrm{dx}$ is equal to

• $- \frac{{\left(1 + {\mathrm{x}}^4\right)}^{{\displaystyle \frac{3}{4}}}}{\mathrm{x}} + \mathrm{C}$

• $- \frac{{\left(1 + {\mathrm{x}}^4\right)}^{{\displaystyle \frac{1}{4}}}}{2\mathrm{x}} + \mathrm{C}$

• $- \frac{{\left(1 + {\mathrm{x}}^4\right)}^{{\displaystyle \frac{1}{4}}}}{\mathrm{x}} + \mathrm{C}$

• $- \frac{{\left(1 + {\mathrm{x}}^4\right)}^{{\displaystyle \frac{1}{4}}}}{{\mathrm{x}}^2} + \mathrm{C}$

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

• $- \cot \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{C}$

• $\tan \left({\mathrm{xe}}^{\mathrm{x}}\right) + \mathrm{C}$

• $\tan \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{C}$

• $- \cot \left({\mathrm{xe}}^{\mathrm{x}}\right) + \mathrm{C}$

$\int \frac{{\mathrm{xe}}^{\mathrm{x}}}{\left(1 + {\mathrm{x}}^2\right)}\mathrm{dx}$ is equal to

• $\frac{- {\mathrm{e}}^{\mathrm{x}}}{\mathrm{x} + 1} + \mathrm{C}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x} + 1} + \mathrm{C}$

• $\frac{{\mathrm{xe}}^{\mathrm{x}}}{\mathrm{x} + 1} + \mathrm{C}$

• $\frac{- {\mathrm{xe}}^{\mathrm{x}}}{\mathrm{x} + 1} + \mathrm{C}$

$\int {\mathrm{e}}^{\mathrm{x}}\left(\sin \left(\mathrm{x}\right) + 2\cos \left(\mathrm{x}\right)\right)\sin \left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• ${\mathrm{e}}^{\mathrm{x}}\cos \left(\mathrm{x}\right) + \mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right) + \mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}{\sin }^2\left(\mathrm{x}\right) + \mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\sin \left(2\mathrm{x}\right) + \mathrm{C}$

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

• $2\sin \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• $\sqrt{2}\sin \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

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

• $2\sqrt{2}\sin \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

$\int \frac{\sqrt{{\mathrm{x}}^2 - 1}}{\mathrm{x}}\mathrm{dx}$ is equal to

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

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

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

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