${\int }_0^{\raisebox{1ex}{$\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

• $\frac{\sqrt{3}}{2}$

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

• $\frac{1}{\sqrt{3}}$

The integral ${\int }_1^{10}{\mathrm{x}}^3\mathrm{dx}$ is approximately evaluated by Trapezoidal rule ${\int }_1^{10}{\mathrm{x}}^3\mathrm{dx}$ = $3\left[\frac{1 + {10}^3}{2} + \mathrm{\alpha } + 7^3\right]$ for  n = 3, then the value of $\mathrm{\alpha }$ is

• 4³

• 4²

• 5³

• None of these

577.

Let f(x) = ${\int }_{\mathrm{u}\left(\mathrm{x}\right)}^{\mathrm{v}\left(\mathrm{x}\right)}\mathrm{g}\left(\mathrm{t}\right)\mathrm{dt}$. Then, f'(x) is equal to

• g[v(x)] - g[u(x)]

• g'[v(x)] - g'[u(x)]

• g[v(x)] v'(x) - g[u(x)] u'(x)

• None of the above

The value of ${\int }_0^{\mathrm{\pi }}\left|\cos \left(\mathrm{x}\right)\right|\mathrm{dx}$ is

• 1

• 2

• 0

• 3

$\int \frac{\left(\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right)}{\sqrt{\left(1 + \sin \left(2\mathrm{x}\right)\right)}}\mathrm{dx}$ equals

• $\log \left(\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right)$

• x

• $\log \left(\sin \left(\cos \left(\mathrm{x}\right)\right)\right)$

• $\log \left(\mathrm{x}\right)$

Using trapezoidal rule and taking n = 4, the value of ${\int }_0^2\frac{\mathrm{dx}}{1 + \mathrm{x}}$ will be

• 1.1167

• 1.1176

• 1.118

• None of these