TITLE: Arbitrary power integration problem
QUESTION [5 upvotes]: How would I solve the following problem.Using u substituion.
$$\int\frac{\log_5(x)}{x} \, dx$$
I am know $\log_5(x)$ is the same as $\frac{\ln(x)}{\ln(5)}$ But how would I proceed.

REPLY [6 votes]: $$\int \frac{\log_5{x}}{x}\,dx = \int \frac{\ln x}{x\ln 5}\,dx = \frac{1}{\ln 5} \int  \underbrace{\ln{x}}_{\large u}\cdot \underbrace{\frac 1x\,dx}_{\large \,du} = \frac1{2\ln 5}(\ln{x})^2 + C$$