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

862.

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

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

• $\frac{1}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\frac{{\mathrm{x}}^{\mathrm{n}}}{{\mathrm{x}}^{\mathrm{n}} + 1}\right) + \mathrm{C}$

• $- \frac{1}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\frac{{\mathrm{x}}^{\mathrm{n}} + 1}{{\mathrm{x}}^{\mathrm{n}}}\right) + \mathrm{C}$

• $\frac{1}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\frac{{\mathrm{x}}^{\mathrm{n}}}{{\mathrm{x}}^{\mathrm{n}} + 1}\right) + \mathrm{C}$

• None of the above

Evaluate $\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^5 + 2\right)}$

• $\frac{1}{10}\log \left|\frac{{\mathrm{x}}^5 + 1}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

• $\frac{1}{5}\log \left|\frac{{\mathrm{x}}^5}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

• $\frac{1}{10}\log \left|\frac{{\mathrm{x}}^5}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

• $\frac{1}{10}\log \left|\frac{{\mathrm{x}}^5 + 1}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

${\int }_2^3\frac{\sqrt{\mathrm{x}}}{\sqrt{5 - \mathrm{x}} + \sqrt{\mathrm{x}}}\mathrm{dx}$ is equal to

• $\frac{1}{4}$

• 1

• $\frac{3}{2}$

• $\frac{1}{2}$

$\mathrm{I} = {\int }_{- 1}^1\left(\frac{{\mathrm{x}}^2 + \sin \left(\mathrm{x}\right)}{1 + {\mathrm{x}}^2}\right)\mathrm{dx}$

• 0

• $2 + \frac{\mathrm{\pi }}{2}$

• $2 - \frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} - 2$

A curve is drawn to pass through the points given by the following table.

x	1	1.5	2	2.5	3	3.5	4
y	2	2.4	2.7	2.8	3	2.6	2.1

Using Simpson's 1/3rd rule, estimate the area bounded by the curve, the x-axis and the lines x = 1, x = 4

• 7.47 sq units

• 7.76 sq units

• 7.78 sq units

• 7.82 sq units