Question

How to integrate -2x^3*e^x^2 dx?

Answer 1

\[\int\limits_{}^{}-2x^3e ^{x^2} dx\]

Answer 2

@jim_thompson5910

Answer 3

e^(x^2) has no elementary antiderivative
so we cannot integrate e^(x^2) and get a known function type (eg: radical, trig, etc)

but -2x^3 can be integrated just fine

Answer 4

use integration by parts

let
u = e^(x^2)
so that means
du = 2x*e^(x^2)dx

and let
dv = -2x^3
integrating both sides gives
v = (-x^4)/2

Answer 5

just in case you didn't know (jim didn't explicitly say it), the derivative of e^u=u' * e^u

Answer 6

\[x^3e^{x^2}=x^2xe^{x^2}\]

Answer 7

\[\Large \int\limits\limits_{}^{}-2x^3e ^{x^2} dx = - \int\limits x^2*2x e^{x^2} dx\]

Answer 8

doing it by parts is messy, but we can use this trick to make things a bit simpler

-2x^3*e^(x^2)
-2x*x^2*e^(x^2)
then we can use u-sub
u = x^2
du = 2xdx
so we'll end up with
-u*e^(u)du

now integration by parts shouldn't be so bad

Answer 9

Not so bad by using this trick. Thank you!

Answer 10

no problem