\[\int \frac{\ln{x}}{{x}^{5}} \, dx\]
1
Use Integration by Parts on ∫ln⁡xx5 dx\int \frac{\ln{x}}{{x}^{5}} \, dx.
Let u=ln⁡xu=\ln{x}, dv=1x5dv=\frac{1}{{x}^{5}}, du=1x dxdu=\frac{1}{x} \, dx, v=−14x4v=-\frac{1}{4{x}^{4}}

2
Substitute the above into uv−∫v duuv-\int v \, du.
−ln⁡x4x4−∫−14x5 dx-\frac{\ln{x}}{4{x}^{4}}-\int -\frac{1}{4{x}^{5}} \, dx

3
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
−ln⁡x4x4+14∫1x5 dx-\frac{\ln{x}}{4{x}^{4}}+\frac{1}{4}\int \frac{1}{{x}^{5}} \, dx

4
Use Power Rule: ∫xn dx=xn+1n+1+C\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C.
−ln⁡x4x4−116x4-\frac{\ln{x}}{4{x}^{4}}-\frac{1}{16{x}^{4}}

5
Add constant.
−ln⁡x4x4−116x4+C-\frac{\ln{x}}{4{x}^{4}}-\frac{1}{16{x}^{4}}+C

Done
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