Mathematics : Integrals

Primitive of cos^-1(x) w.r.t. x is

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

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

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

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

If y = log(cot(x)), then ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{ydx}$ is equal to

• 1

• 0

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

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

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

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

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

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

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

   

Primitive of $\frac{1}{4\sqrt{\mathrm{x}} + \mathrm{x}}$ is equal to

• $2\log \left|1 + 4\sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{2}\log \left|4 - \sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

• $2\log \left| 4 + \sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{2}\log \left| 4 + \sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

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

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

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

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

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} +\hspace{0.17em}\mathrm{c}$

${\int }_0^1\frac{{\mathrm{x}}^2}{1 + {\mathrm{x}}^2}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{4} - 1$

• $1 - \frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} - 1$

• $1 - \frac{\mathrm{\pi }}{4}$

Minimize : z = 3x + y, subject to 2x + 3y $\le$ 6, x + y $\ge$ 1, $\mathrm{x} \ge 0, \mathrm{y} \ge 0$

• x = 1, y = 1

• x = 0, y = 1

• x = 1, y = 0

• x = - 1, y = - 1

$\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^7 + 1\right)}$ is equal to

• $\log \left(\frac{{\mathrm{x}}^7}{{\mathrm{x}}^7 + 1}\right)$

• $\frac{1}{7}\log \left(\frac{{\mathrm{x}}^7}{{\mathrm{x}}^7 + 1}\right) + \mathrm{c}$

• $\log \left(\frac{{\mathrm{x}}^{7 }+ 1}{{\mathrm{x}}^7}\right) + \mathrm{c}$

• $\frac{1}{7}\log \left(\frac{{\mathrm{x}}^7 +\hspace{0.17em}1}{{\mathrm{x}}^7}\right)$

Using Trapezoidal rule and following table ${\int }_0^8\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

x	0	0	4	6	8
f(x)	2	5	10	17	26

• 184

• 92

• 46

• - 36

$\int \frac{\mathrm{dx}}{\mathrm{x} +\hspace{0.17em}\sqrt{\mathrm{x}}}$ is equal to

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

• $2\log \left(1 +\hspace{0.17em}\sqrt{\mathrm{x}}\right) + \mathrm{c}$

• $\frac{1}{4}\log \left(1 +\hspace{0.17em}\sqrt{\mathrm{x}}\right) + \mathrm{c}$

• $3\log \left(1 +\hspace{0.17em}\sqrt{\mathrm{x}}\right) + \mathrm{c}$