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

• $\frac{1}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\frac{{\mathrm{x}}^{\mathrm{n}}}{{\mathrm{x}}^{\mathrm{n}} + 1}\right) + \mathrm{C}$

• $- \frac{1}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\frac{{\mathrm{x}}^{\mathrm{n}} + 1}{{\mathrm{x}}^{\mathrm{n}}}\right) + \mathrm{C}$

• $\frac{1}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\frac{{\mathrm{x}}^{\mathrm{n}}}{{\mathrm{x}}^{\mathrm{n}} + 1}\right) + \mathrm{C}$

• None of the above

Evaluate $\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^5 + 2\right)}$

• $\frac{1}{10}\log \left|\frac{{\mathrm{x}}^5 + 1}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

• $\frac{1}{5}\log \left|\frac{{\mathrm{x}}^5}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

• $\frac{1}{10}\log \left|\frac{{\mathrm{x}}^5}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

• $\frac{1}{10}\log \left|\frac{{\mathrm{x}}^5 + 1}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

${\int }_2^3\frac{\sqrt{\mathrm{x}}}{\sqrt{5 - \mathrm{x}} + \sqrt{\mathrm{x}}}\mathrm{dx}$ is equal to

• $\frac{1}{4}$

• 1

• $\frac{3}{2}$

• $\frac{1}{2}$

$\mathrm{I} = {\int }_{- 1}^1\left(\frac{{\mathrm{x}}^2 + \sin \left(\mathrm{x}\right)}{1 + {\mathrm{x}}^2}\right)\mathrm{dx}$

• 0

• $2 + \frac{\mathrm{\pi }}{2}$

• $2 - \frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} - 2$

A curve is drawn to pass through the points given by the following table.

x	1	1.5	2	2.5	3	3.5	4
y	2	2.4	2.7	2.8	3	2.6	2.1

Using Simpson's 1/3rd rule, estimate the area bounded by the curve, the x-axis and the lines x = 1, x = 4

• 7.47 sq units

• 7.76 sq units

• 7.78 sq units

• 7.82 sq units

Which of these methods for numerical integration is also called as parabolic formula?

• Simpson's one-third rule

• Simpson's three-eighth's rule

• Trapezoidal rule

• None of the above

587.

Calculate by Trapezoidal rule an approximate value of ${\int }_{- 3}^3{\mathrm{x}}^4\mathrm{dx}$ by taking seven equidistant ordinates

• 98

• 97.2

• 100

• 115

The value of ${\int }_0^1{\sin }^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} - 1$

• $\frac{\mathrm{\pi }}{2} + 1$

• 0

By Simposon's rule taking n = 4, the value of the integral ${\int }_0^1\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}$ is equal to

• 0.785

• 0.788

• 0.781

• None of these

Thevalue of integral ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(\sqrt{\tan \left(\mathrm{x}\right)} + \sqrt{\cot \left(\mathrm{x}\right)}\right)\mathrm{dx}$ is

• $\sqrt{2}\mathrm{\pi }$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• 0