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
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