If `int_0^1e^ (-x^2) dx=a ,` then find the value of `int_0^1x^2e^ (-x^2) dx` in terms of `a`.

Explanation

`I=int_(0)^(1)x^(2)e^(-x^(2))dx=(-1)/2int_(0)^(1)x(-2x)e^(-x^(2))dx`
`=-1/2(xe^(-x^(2))|_(0)^(1)-int_(0)^(1)e^(-x^(2))dx)` (Integrating by parts)
`=-1/(2e)+1/2 a`