The value of $\int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin ^2 x}{1+2^x} d x$ is

The area (in sq. units) of the region described by $\left\{(x, y) / y^2 \leq 2 x\right.$ and $\left.y \geq(4 x-1)\right\}$ is

Let $y=y(x)$ be the solution of the differential equation $(x \log x) \frac{d y}{d x}+y=2 x \log x(x \geq 1)$ then $y(\mathrm{e})$ is equal to

The set of all points, for which $f(x)=x^2 e^{-x}$ strictly increases, is