Mathematics : Integrals

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

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

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

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