Question

A particle moves along the curve 6y = x³ + 2. Find the points on the curve at which y-coordinate is changing 2 times as fast as x-coordinate.

Find : $\textstyle\int e^{x}\,{\frac{\sqrt{1+\sin2x}}{1+\cos2\,x}}\,d x$.

Find : $\textstyle\int$$\frac{d x}{\sqrt{3-2\,x-x^{2}}}$

Verify that ax² + by² = 1 is a solution of differential equation $x(y y_{2}+y_{1}^{2})=y y_{1}.$

Find : $\textstyle\int$$\frac{1}{\sqrt{x^{2}-4\,x}}\,d x$

The radius r of a right circular cylinder is decreasing at the rate of 3 cm/min. and its height h is increasing at the rate of 2 cm/min. When r =7 cm and h = 2 cm, find the rate of change of the volume of cylinder. $[\mathrm{Use}\:\pi=\:\frac{22}{7}\:]$

Find the general solution of the differential equation ${\frac{d y}{d x}}+{\frac{2}{x}}y=x$