integral of x^2*(exp(-x))

Question

will integration by parts be used to solved this problem?

Answer 1

Let I=∫x²e^-xdx.

Let u=x², du=2xdx; dv=e^-xdx, v=-e^-x.

I=∫udv=uv-∫vdu=-x²e^-x+2∫xe^-xdx=-x²e^-x+2J, where J=∫xe^-xdx.

Let u=x, du=dx.

J=uv-∫vdu=-xe^-x+∫e^-xdx=-xe^-x-e^-x.

I=-x²e^-x-2xe^-x-2e^-x=-e^-x(x²+2x+2).

CHECK

dI/dx=e^-x[(x²+2x+2)-(2x+2)]=x²e^-x✔️