Question:
Evaluate $\int \frac{\log x}{x^{3}} d x$
Solution:
Use method of integration by parts
$y=\log x \int \frac{1}{x^{3}} d x-\int \frac{d}{d x} \log x\left(\int \frac{1}{x^{3}} d x\right) d x$
$y=-\log x \frac{1}{2 x^{2}}+\int \frac{1}{2 x^{3}} d x$
$y=-\frac{1}{2 x^{2}} \log x-\frac{1}{4 x^{2}}+c$
$y=-\frac{1}{4 x^{2}}(2 \log x+1)+c$