Question

Integrate the function: $x^2e^x$

Answer 1

Let I = ∫x2ex dx

Taking $x^2$ as first function and ex as second function and integrating by parts, we obtain

I= x2∫ex dx - ∫{$(\frac {d}{dx}x^2)$∫exdx} dx

I = x2ex - ∫2x.exdx

I = x2ex-2∫x.ex dx

Again integrating by parts,we obtain

I =x2ex - 2[x.∫exdx - ∫{$(\frac {d}{dx}x)$.∫exdx} dx]

I = x2ex - 2[xex-∫exdx]

I = x2ex - 2[xex - ex]

I = x2ex - 2xex + 2ex + C

I = ex(x2 - 2x + 2) + C