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

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

Find : $\textstyle\int$${\frac{e^{x}(x-3)}{(x-1)^{3}}}d x\,.$

Find : $\textstyle\int$${\frac{\left(x^{4}-x\right)^{\frac{1}{4}}}{x^{5}}}\,.$

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

Find the particular solution of the differential equation ${\frac{d y}{d x}}={\frac{1+y^{2}}{1+x^{2}}}$ given that y(0) = ${\sqrt{3}}.$

If A and B are two independent events, then prove that the probability of occurrence of at least one of A and B is given by 1 – P(A’) · P(B’)

If P(A) = 0.4, P(B) = p, P(A U B) = 0.6 and A and B are given to the independent events, find the value of ‘p’.

A couple has 2 children. Find the probability that both are boys, if it is known that (i) one of them is a boy (ii) the older child is a boy.

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the event “number obtained is even” and B be the event “number obtained is red”. Find if A and B 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.

If A and B are two events such that P(A) = 0.4, P(B) = 0.8 and P(B/A) = 0.6, then find P(A/B).

If P(A) = 2/5, P(B) = 1/3, P(A ∩ B) = 1/5, then find P(A' / B') .

Prove that if E and F are independent events, then the events E and F' are also independent.

A box contains 50 bolts and 50 nuts. Half of the bolts and nuts are rusted. If two items are drawn with replacement, what is the probability that either both are rusted or both are bolts.