The integral $$\int\limits_1^e {\left\{ {{{\left( {{x \over e}} \right)}^{2x}} - {{\left( {{e \over x}} \right)}^x}} \right\}} \,$$ log_e x dx is equal to :

The integral $$\int {{{3{x^{13}} + 2{x^{11}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^4}}}} \,dx$$ is equal to : (where C is a constant of integration)

Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $$\Delta $$S'BS is a right angled triangle with right angle at B and area ($$\Delta $$S'BS) = 8 sq. units, then the length of a latus rectum of the ellipse is :

Let S be the set of all real values of $$\lambda $$ such that a plane passing through the points (–$$\lambda $$², 1, 1), (1, –$$\lambda $$², 1) and (1, 1, – $$\lambda $$²) also passes through the point (–1, –1, 1). Then S is equal to :