Question:
Evaluate $\int \log _{10} \mathrm{x} \mathrm{dx}$
Solution:
Use the method of integration by parts
$y=\int 1 \times \log _{10} x d x$
$y=\log _{10} x \int d x-\int \frac{d}{d x} \log _{10} x\left(\int d x\right) d x$
$y=x \log _{10} x-\int x \frac{1}{x \log _{e} 10} d x$
$y=x \log _{10} x-\frac{x}{\log _{e} 10}+c$
$y=x\left(\log _{e} x-1\right) \log _{10} e+c$