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(x^2*ln(x))/2-x^2/4+C