How do I properly integrate (x^3)(e^(x^2))?

[johann1301]

Homework Statement

Use both substitution and integration by parts to solve:

∫x³e^x2dx

2. The attempt at a solution

∫x³e^x2dx

∫e^x2x³dx

(1/2x)e^x2x³-∫(1/2x)e^x23x²dx

(1/2)e^x2x²-(3/2)∫e^x2xdx

(e^x2x²/2)-(3/2)∫e^ux(du/2x)

(e^x2x²/2)-(3/2)(1/2)∫e^udu

(e^x2x²/2)-(3/4)e^u + C

(e^x2x²/2)-(3/4)e^x2 + C

The solution is supposed to be; (e^x2x²/2)-(1/2)e^x2 + C

Where am i going wrong?

 

[D H]

You assumed that ∫e^x2dx = e^x2/(2x). That's not true.

You do not want to take the route of trying to integrate e^x2. No matter how hard you try you will not be able to find that integral. This integral cannot be expressed in terms of elementary functions. You need to take another approach.

Hint: That u-substitution you used to solve ∫xe^x2dx might come in handy if you use it earlier on.

 

[johann1301]

Solved it. Thanks a lot!