# Problem of the Week

## Updated at Dec 16, 2013 3:55 PM

This week's problem comes from the calculus category.

How would you integrate $$\frac{\ln{x}}{{x}^{5}}$$?

Let's begin!

$\int \frac{\ln{x}}{{x}^{5}} \, dx$

 1 Use Integration by Parts on $$\int \frac{\ln{x}}{{x}^{5}} \, dx$$.Let $$u=\ln{x}$$, $$dv=\frac{1}{{x}^{5}}$$, $$du=\frac{1}{x} \, dx$$, $$v=-\frac{1}{4{x}^{4}}$$2 Substitute the above into $$uv-\int v \, du$$.$-\frac{\ln{x}}{4{x}^{4}}-\int -\frac{1}{4{x}^{5}} \, dx$3 Use Constant Factor Rule: $$\int cf(x) \, dx=c\int f(x) \, dx$$.$-\frac{\ln{x}}{4{x}^{4}}+\frac{1}{4}\int \frac{1}{{x}^{5}} \, dx$4 Use Power Rule: $$\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C$$.$-\frac{\ln{x}}{4{x}^{4}}-\frac{1}{16{x}^{4}}$5 Add constant.$-\frac{\ln{x}}{4{x}^{4}}-\frac{1}{16{x}^{4}}+C$Done-ln(x)/(4*x^4)-1/(16*x^4)+C