The value of ${\int }_0^1\mathrm{xdx}$ by Trapezoidal rule taking x = 4 is

• 0.34375

• 0.5

• 0.38387

• 0.353367

$\int \frac{\mathrm{dx}}{{\mathrm{x}}^4 - 1}$ is equal to

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

• $\log \left|\frac{\mathrm{x} - 1}{\mathrm{x} + 1}\right| + \mathrm{C}$

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

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

If e⁰ = 1, e¹ = 2.72, e² = 7.39, e³ = 20.09, e⁴ = 54.60, then the value of ${\int }_0^4{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ using Simpson's rule, will be

• 5.387

• 53.87

• 52.78

• 53.17

594.

${\int }_{- 1}^2\left|{\mathrm{x}}^3 - \mathrm{x}\right|\mathrm{dx}$ is equal to

• 11

• 4

• $\frac{11}{4}$

• $\frac{{\displaystyle 4}}{{\displaystyle 11}}$

According to Simpson's rule, the value of ${\int }_1^7\frac{\mathrm{dx}}{\mathrm{x}}$ is

• 1.358

• 1.958

• 1.625

• 1.458

If $\int \sin \left\{2{\tan }^{-1}\left(\sqrt{\frac{1 - \mathrm{x}}{1 + \mathrm{x}}}\right)\right\}\mathrm{dx} = {\mathrm{Asin}}^{-1}\left(\mathrm{x}\right) + \mathrm{Bx}\sqrt{1 - {\mathrm{x}}^2} + \mathrm{C}$, then A + B is equal to

• 10

• $\frac{1}{2}$

• 1

• $- \frac{1}{2}$

${\int }_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{e}$}\right.}^1\left|\log \left(\mathrm{x}\right)\right|\mathrm{dx}$ is equal to

• $\frac{1}{\mathrm{e}}$

• e

• $2\left(1 - \frac{1}{\mathrm{e}}\right)$

• None of the above

$\underset{\mathrm{x}\to 0}{\lim }\frac{{\int }_0^{{\mathrm{x}}^2}\sin \left(\sqrt{\mathrm{t}}\right)\mathrm{dt}}{{\mathrm{x}}^3}$ is equal to

• $\frac{2}{3}$

• $\frac{1}{3}$

• 0

• $\infty$

By trapezoidal rule, the approximate value of the integral ${\int }_0^6\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}$ is

• 1.3128

• 1.4108

• 1.4218

• None of these

The value of the integral $\mathrm{I} = \int \left(\sqrt{\tan \left(\mathrm{x}\right)} + \sqrt{\cot \left(\mathrm{x}\right)}\right)\mathrm{dx}, \mathrm{where} \mathrm{x} \in \left(0, \frac{\mathrm{\pi }}{2}\right)$, is

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

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

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

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