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

• $2 + \frac{\mathrm{\pi }}{2}$

• $2 - \frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} - 2$

Find the area of the region {(x, y) : x² $\le \mathrm{y} \le \left|\mathrm{x}\right|$}

• $\frac{1}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{4}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{2}{3} \mathrm{sq} \mathrm{unit}$

• None of these

Determine the area included between the curve $\mathrm{y} {=}^2 \cos \left(\mathrm{x}\right), 0 \le \mathrm{x} \le \frac{\mathrm{\pi }}{2}$ and the axes.

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

The differential equation whose solution represents the family y = ae^3x + be^x is given by

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 4\frac{\mathrm{dy}}{\mathrm{dx}} - 3\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 4\frac{\mathrm{dy}}{\mathrm{dx}} - 3\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 4\frac{\mathrm{dy}}{\mathrm{dx}} + 3\mathrm{y} = 0$

• None of the above

Solve $\frac{2\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{y}}{\mathrm{x}} + \frac{\mathrm{y}}{{\mathrm{x}}^2}$

• y = x + $\mathrm{C}\sqrt{\mathrm{xy}}$

• $\mathrm{y} = \mathrm{x} - \mathrm{C}\sqrt{\mathrm{xy}}$

• $\mathrm{y} = \mathrm{x} + \mathrm{Cy}\sqrt{\mathrm{x}}$

• $\mathrm{y} = \mathrm{x} + \mathrm{C}\sqrt{\mathrm{y}}$

47.

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

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

Solve $\left(\mathrm{x} +\hspace{0.17em}2{\mathrm{y}}^3\right)\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y}, \mathrm{y} >\hspace{0.17em}0$

• y = x³ + Cy

• x = y³ + Cy

• y = x³ - Cy

• x = y³ - Cy