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

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

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

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

• None of the above

$\int \frac{\log \left(\mathrm{x} + \sqrt{1 + {\mathrm{x}}^2}\right)}{\sqrt{1 + {\mathrm{x}}^2}}\mathrm{dx}$ is equal to

• $\frac{1}{2}{\left[\log \left(\mathrm{x} + \sqrt{1 + {\mathrm{x}}^2}\right)\right]}^2 + \mathrm{c}$

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

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

• None of the above

$\int \frac{{\sin }^8\left(\mathrm{x}\right) - {\cos }^8\left(\mathrm{x}\right)}{1 - 2{\sin }^2\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

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

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

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

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

$\int \frac{\mathrm{dx}}{\left(\sin \left(\mathrm{x}\right) + \sin \left(2\mathrm{x}\right)\right)}$ is equal to

• $\frac{1}{6}\log \left(1 - \cos \left(\mathrm{x}\right)\right) + \frac{1}{2}\log \left(\log \left(1 + \cos \left(\mathrm{x}\right)\right)\right) - \frac{2}{3}\log \left(1 + 2\cos \left(\mathrm{x}\right)\right) + \mathrm{c}$

• $6\log \left(1 - \cos \left(\mathrm{x}\right)\right) + 2\log \left(\log \left(1 + \cos \left(\mathrm{x}\right)\right)\right) - \frac{2}{3}\log \left(1 + 2\cos \left(\mathrm{x}\right)\right) + \mathrm{c}$

• $6\log \left(1 - \cos \left(\mathrm{x}\right)\right) + \frac{1}{2}\log \left(\log \left(1 + \cos \left(\mathrm{x}\right)\right)\right) + \frac{2}{3}\log \left(1 + 2\cos \left(\mathrm{x}\right)\right) + \mathrm{c}$

• None of the above

565.

$\int \left[\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\text{'}\text{'}\left(\mathrm{x}\right) - \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)\mathrm{g}\left(\left(\mathrm{x}\right)\right)\right]\mathrm{dx}$ is equal to

• $\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\text{'}\left(\mathrm{x}\right)}$

• f'(x)g(x) - f(x)g'(x)

• f(x)g'(x) - f'(x)g(x)

• f(x)g'(x) + f'(x)g(x)

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

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