How do you evaluate the integral ∫x2ex?

1 Answer
Jacob T.
Jun 1, 2017

∫x2exdx=ex(x2−2x+2)+C

Explanation:

∫x2exdx
=x2⋅ex−∫2x⋅ex
=x2⋅ex−(2⋅x⋅ex−∫2⋅ex)
=x2⋅ex−(2⋅x⋅ex−2⋅ex)+C
=ex(x2−2x+2)+C

Explanation:
Integrate by parts :
∫x2exdx
Let u=x2, v=ex. You will get (dv)=ex⋅dx, (du)=2x⋅dx, .

To reduce the power of x in the integral, you may rewrite it in this form ∫u⋅dv. This new expression is equivalent to the original one. (by replacing x2 with u and ex⋅dx with (dv).)

∫u⋅dv=u⋅v−∫du⋅v
∫x2exdx=x2⋅ex−∫2x⋅ex

Now, let's do the partial integration again with m=2⋅x this time. In this case dm=2⋅dx

∫m⋅dv=m⋅v−∫dm⋅v
∫2x⋅ex=2⋅x⋅ex−∫2⋅ex=2⋅x⋅ex−2⋅ex+C
(include the arbritary constant C in the expression)

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