21.

Evaluate: ${\int }_0^{\mathrm{\pi }}\frac{\mathrm{xsinx}}{1 + {\cos }^2\mathrm{x}} \mathrm{dx}$

Solve the following differential equation:
(x² − y²⁾ dx + 2xy dy = 0   given that y = 1 when x = 1

Solve the following differential equation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} ( 2\mathrm{y} - \mathrm{x} )}{\mathrm{x} ( 2\mathrm{y} + \mathrm{x})}$,   if y = 1 when x = 1

Solve the following differential equation:

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

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{j}} - \hat{\mathrm{k}}, \mathrm{find} \mathrm{a} \mathrm{vector} \overrightarrow{\mathrm{c}} \mathrm{such} \mathrm{that} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}} = \overrightarrow{\mathrm{b}} \\ \mathrm{and} \overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{c}} = 3 \end{gathered}$$

If $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = 0 \mathrm{and} \left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 5 \mathrm{and} \left|\overrightarrow{\mathrm{c}}\right| = 7,$ show that the angle between $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{b}$ is 60⁰.

Find the shortest distance between the following lines:

$\frac{\mathrm{x} - 3}{1} = \frac{\mathrm{y} - 5}{-2} = \frac{\mathrm{z} - 7}{1} \mathrm{and} \frac{{\displaystyle \mathrm{x} + 1}}{{\displaystyle 7}} = \frac{{\displaystyle \mathrm{y} + 1}}{{\displaystyle -6}} = \frac{{\displaystyle \mathrm{z} + 1}}{{\displaystyle 1}}$

Find the point on the line $\frac{\mathrm{x} + 2}{3} = \frac{\mathrm{y} + 1}{2} = \frac{\mathrm{z} - 3}{2}$ at a distance 3 $\sqrt{2}$ from the point
(1, 2, 3).

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes.

Using integration find the area of the region bounded by the parabola y² = 4x and the circle 4x² + 4y² = 9.