Correct value of ${\int }_0^{\mathrm{\pi }}\left|{\sin }^4\left(\mathrm{x}\right)\right|\mathrm{dx}$ is

• $\frac{8\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{4\mathrm{\pi }}{3}$

• $\frac{3\mathrm{\pi }}{8}$

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

• $- \frac{\mathrm{\pi }}{2}\log \left(2\right)$

• $\mathrm{\pi log}\left(\frac{1}{2}\right)$

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

• $\log \left(2\right)$

${\int }_0^{\mathrm{\pi }}{\cos }^3\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• - 1

• 0

• 1$\mathrm{\pi }$

• 1

854.

Suppose f is such that f(- x) = - f(x), for every x and ${\int }_0^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = 5$, then ${\int }_{- 1}^0\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$ is equal to

• 10

• 5

• 0

• - 5

Withthehelp of Trapezoidal rule fornumerical integration and the following table

x	0	0.25	0.50	0.75	1
f(x)	0	0.625	0.2500	0.5625	1

the value of ${\int }_0^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is

• 0.35342

• 0.34375

• 0.34457

• 0.33334

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\mathrm{xsin}\left(\mathrm{x}\right) . \cos \left(\mathrm{x}\right)}{{\cos }^4\left(\mathrm{x}\right) + {\sin }^4\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{{\mathrm{\pi }}^2}{8}$

• $\frac{{\mathrm{\pi }}^2}{16}$

• 1

• 0

Use Simpson's $\frac{1}{3}$ rule to find the value of ${\int }_1^5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ given,

x	1	2	3	4	5
y	10	50	70	80	100

• 140.88

• 256.66

• 160.26

• None of these

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

• $\mathrm{xlog}\left(1 + \cos \left(\mathrm{x}\right)\right) + \mathrm{c}$

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

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

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

The value of $\int \frac{\cos \left(\sqrt{\mathrm{x}}\right)}{\sqrt{\mathrm{x}}}\mathrm{dx}$ will be

• $2\sin \left(\sqrt{\mathrm{x}}\right) + \mathrm{c}$

• $2\cos \left(\sqrt{\mathrm{x}}\right) + \mathrm{c}$

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

• $\sqrt{2}\sin \left(\mathrm{x}\right) + \mathrm{c}$

The value of ${\int }_2^3\frac{\sqrt{\mathrm{x}}}{\sqrt{5 - \mathrm{x}} + \sqrt{\mathrm{x}}}\mathrm{dx}$ will be

• $\frac{\sqrt{3}}{2}$

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

• $\frac{1}{\sqrt{3}}$