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

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

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

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

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

   

The value of $\int \frac{{\mathrm{x}}^2}{1 +\hspace{0.17em}{\mathrm{x}}^6}\mathrm{dx}$ is

• x³ + c

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

• log(1 + x³) + c

• None of these

$\int \left(1 - \cos \left(\mathrm{x}\right)\right){\csc }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

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

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

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

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

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

• $\sin \left(\frac{\mathrm{x}}{2}\right) - \cos \left(\frac{\mathrm{x}}{2}\right)$

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

• $2\left[\sin \left(\frac{\mathrm{x}}{2}\right) - \cos \left(\frac{\mathrm{x}}{2}\right)\right] +\hspace{0.17em}\mathrm{c}$

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

${\int }_1^3\frac{\cos \left(\log \left(\mathrm{x}\right)\right)}{\mathrm{x}}\mathrm{dx}$ is equal to

• 1

• $\cos \left(\log \left(3\right)\right)$

• $\sin \left(\log \left(3\right)\right)$

• $\frac{\mathrm{\pi }}{4}$

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

• $- \frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$