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.
\[\frac{{e}^{{x}^{2}}}{2}+C\]

Done
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