Problem of the Week

Updated at Mar 3, 2014 2:23 PM

This week we have another calculus problem:

How can we solve for the integral of \(x\ln{x}\)?

Let's start!



\[\int x\ln{x} \, dx\]

1
Use Integration by Parts on \(\int x\ln{x} \, dx\).
Let \(u=\ln{x}\), \(dv=x\), \(du=\frac{1}{x} \, dx\), \(v=\frac{{x}^{2}}{2}\)

2
Substitute the above into \(uv-\int v \, du\).
\[\frac{{x}^{2}\ln{x}}{2}-\int \frac{x}{2} \, dx\]

3
Use Constant Factor Rule: \(\int cf(x) \, dx=c\int f(x) \, dx\).
\[\frac{{x}^{2}\ln{x}}{2}-\frac{1}{2}\int x \, dx\]

4
Use Power Rule: \(\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C\).
\[\frac{{x}^{2}\ln{x}}{2}-\frac{{x}^{2}}{4}\]

5
Add constant.
\[\frac{{x}^{2}\ln{x}}{2}-\frac{{x}^{2}}{4}+C\]

Done
Cymath foriOSAndroid