\[\int x\ln{x} \, dx\]
1
Use Integration by Parts on ∫xln⁡x dx\int x\ln{x} \, dx.
Let u=ln⁡xu=\ln{x}, dv=xdv=x, du=1x dxdu=\frac{1}{x} \, dx, v=x22v=\frac{{x}^{2}}{2}

2
Substitute the above into uv−∫v duuv-\int v \, du.
x2ln⁡x2−∫x2 dx\frac{{x}^{2}\ln{x}}{2}-\int \frac{x}{2} \, dx

3
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
x2ln⁡x2−12∫x dx\frac{{x}^{2}\ln{x}}{2}-\frac{1}{2}\int x \, dx

4
Use Power Rule: ∫xn dx=xn+1n+1+C\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C.
x2ln⁡x2−x24\frac{{x}^{2}\ln{x}}{2}-\frac{{x}^{2}}{4}

5
Add constant.
x2ln⁡x2−x24+C\frac{{x}^{2}\ln{x}}{2}-\frac{{x}^{2}}{4}+C

Done
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