# Problem of the Week

## Updated at Dec 30, 2013 9:58 AM

This week we have another calculus problem:

How would you integrate $$x{e}^{{x}^{2}}$$?

Let's start!

$\int x{e}^{{x}^{2}} \, dx$

 1 Use Integration by Substitution.Let $$u={x}^{2}$$, $$du=2x \, dx$$, then $$x \, dx=\frac{1}{2} \, du$$2 Using $$u$$ and $$du$$ above, rewrite $$\int x{e}^{{x}^{2}} \, dx$$.$\int \frac{{e}^{u}}{2} \, du$3 Use Constant Factor Rule: $$\int cf(x) \, dx=c\int f(x) \, dx$$.$\frac{1}{2}\int {e}^{u} \, du$4 The integral of $${e}^{x}$$ is $${e}^{x}$$.$\frac{{e}^{u}}{2}$5 Substitute $$u={x}^{2}$$ back into the original integral.$\frac{{e}^{{x}^{2}}}{2}$6 Add constant.$\frac{{e}^{{x}^{2}}}{2}+C$Donee^x^2/2+C