How do I find this integral?

Question

x^2*e^x*dx

Answer 1

∫ x² e^x dx

You can use Integration by parts..
∫u dv = uv - ∫v du

note if you let v be e^x and the x^2 be u.. then the du term will drop to a constant after applying integration by parts twice. So let..
u = x^2
du = 2x dx
dv = e^x dx
v = e^x


∫ x² e^x dx = x² e^x - ∫ 2x e^x dx

Integrate the ∫ 2x e^x by parts
u = 2x
du = 2 dx
dv = e^x dx
v = e^x

∫ x² e^x = x² e^x - [2x e^x - ∫ 2 e^x dx]
∫ x² e^x = x² e^x - [2x e^x - 2∫e^x dx]
∫ x² e^x = x² e^x - [2x e^x - 2 e^x] + C
∫ x² e^x = x² e^x - 2x e^x + 2 e^x + C

Answer 2

integral uv' dx = uv - integral u'v dx

Let's take v = e^x, u=x^2

integral x^2 e^x dx = x^2 e^x - integral 2x e^x dx =
= x^2 e^x - (2x e^x - integral 2e^x dx) =
= x^2 e^x - (2x e^x - 2e^x) + c =
= x^2 e^x - 2x e^x + 2e^x + c

Answer 3

we can do it by Intigration by parts


x^2*e^x - 2*x*e^x + e^x