The value of the integral $$\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx} $$ is

Evaluate : $$\int {{{\left( {x - 1} \right){e^x}} \over {{{\left( {x + 1} \right)}^3}}}dx} $$

Find the coordinates of the point on the curve $$y = {x \over {1 + {x^2}}}$$
where the tangent to the curve has the greatest slope.

Given positive integers $$r > 1,\,n > 2$$ and that the coefficient of $$\left( {3r} \right)$$th and $$\left( {r + 2} \right)$$th terms in the binomial expansion of $${\left( {1 + x} \right)^{2n}}$$ are equal. Then