Mathematics : Integrals

${\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt} = {\mathrm{x}}^2 + {\mathrm{e}}^{\mathrm{x}}$ (x > 0), then f(1) is equal to

• 1 + e

• 2 + e

• 3 + e

• e

$\int \frac{\mathrm{x} + 1}{{\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\mathrm{dx} =$

• - x^3/2 + x^1/2 + C

• x^1/2

• x^3/2 + 2x^1/2 + C

• $\frac{2{\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{3} + 2{\mathrm{x}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + \mathrm{C}$

$\int \frac{\mathrm{dx}}{{\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{- \mathrm{x}} + 2}$ is equal to

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

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

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

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

${\int }_{- 1}^0\frac{\mathrm{dx}}{{\mathrm{x}}^2 + \mathrm{x} + 2}$ is equal to

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

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\mathrm{\pi }$

• $\frac{4}{\sqrt{7}}{\tan }^{-1}\left(\frac{1}{\sqrt{7}}\right)$

${\int }_{- 2}^2\left|1 - {\mathrm{x}}^2\right|\mathrm{dx}$ is equal to :

• 4

• 2

• - 2

• 0

$\int \frac{\sqrt{\tan \left(\mathrm{x}\right)}}{\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to :

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

• $\sqrt{\cot \left(\mathrm{x}\right)} + \mathrm{c}$

• $2\sqrt{\tan \left(\mathrm{x}\right)} + \mathrm{c}$

• ${\tan }^2\left(\mathrm{x}\right) + \mathrm{c}$

$\int \frac{\mathrm{dx}}{{\mathrm{x}}^2 + 4\mathrm{x} + 13}$ is equal to :

• $\log \left({\mathrm{x}}^2 + 4\mathrm{x} + 130\right) + \mathrm{c}$

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

• $\log \left(2\mathrm{x} + 4\right)$ + c

• $\frac{1}{{\mathrm{x}}^2 + 4\mathrm{x} + 13} + \mathrm{c}$

${\int }_0^1\frac{\mathrm{d}}{\mathrm{dx}}\left[{\sin }^{-1}\left(\frac{2\mathrm{x}}{1 + {\mathrm{x}}^2}\right)\right]\mathrm{dx}$ is equal to :

• 0

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

${\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{\sin \left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

$\int \frac{\sin \left(\mathrm{x}\right)}{\sin \left(\mathrm{x} - \mathrm{\alpha }\right)}\mathrm{dx}$ is equal to

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

• $\sin \left(\mathrm{x} - \mathrm{\alpha }\right) + \sin \left(\mathrm{x}\right) + \mathrm{c}$

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

• $\cos \left(\mathrm{x} - \mathrm{\alpha }\right) + \cos \left(\mathrm{x}\right) + \mathrm{c}$