Question

Find the value of $\int \frac{dx}{\log x^x\left[\right(\log x{)}^2-3\log x-10]}$.

Answer 1

Let $I=\int \frac{dx}{log{x}^x\left[\right(logx{)}^2-3logx-10]}$
Put $\log x=t\Rightarrow \frac{1}{x}dx=dt$
$I=\int \frac{1}{t(t^2-3t-10)}dt=\int (-\frac{1}{35(5-t)}-\frac{1}{10t}+\frac{1}{14(t+2)})dt$
$=\frac{1}{35}\log (5-t)-\frac{1}{10}\log t+\frac{1}{14}\log (t+2)$
$=-\frac{1}{10}log\left(logx\right)+\frac{1}{35}log(logx-5)+\frac{1}{14}log(logx+2)$