$$ \mathop {\lim }\limits_{n \to \infty } \frac{1}{n^2}\left[e^{1 / n}+2 e^{2 / n}+3 e^{3 / n}+\ldots+2 n e^2\right]= $$

Let $m, n, p, q$ be four positive integers. If

$$ \begin{aligned} & \int_0^{2 \pi} \sin ^m x \cos ^n x d x=4 \int_0^{\pi / 2} \sin ^m x \cos ^n x d x \int_0^{2 \pi} \sin ^p x \cos ^n x d x=0 \\ & \int_0^\pi \sin ^p x \cos ^q x d x=0, a=m+n+p \text { and } b=m+n+q, \text { then } \end{aligned} $$

The area of the region bounded by the curves $y=x^3, y=x^2$ and the lines $x=0$ and $x=2$ is

The substitution required to reduce the differential equation $t^2 d x+\left(x^2-t x+t^2\right) d t=0$ to a differential equation which can be solved by variables separable method is