Question

Find : $\textstyle\int$$\left({\frac{1-x}{1+x^{2}}}\right)^{2}e^{x}d x~.$

Find : $\textstyle\int$$\frac{x}{\sqrt{32-x^{2}}}\,d x$

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

If E and F be two events such that P(E) = 1/3, P(F) = 1/4, find P(E U F) if E and F are independent events.

How many equivalence relations on the set {1, 2, 3} containing (1, 2) and (2, 1) are there in all ?Justify your answer.

The radius r of the base of a right circular cone is decreasing at the rate of 2 cm/min. and height h is increasing at the rate of 3 cm/min. When r = 3.5 cm and h = 6 cm, find the rate of change of the volume of the cone. $[\mathrm{Use}\:\pi=\:\frac{22}{7}\:]$

If E and F are independent events, then show that (i) E and F' are independent events. (ii) E' and F are also independent events.