Question:
Write a value of $\int \frac{\log \mathrm{x}^{\mathrm{n}}}{\mathrm{x}} \mathrm{dx}$.
Solution:
let $\log x^{n}=t$
Differentiating on both sides we get,
$\frac{1}{x^{n}} n x^{n-1} d x=d t$
$\frac{n}{x} d x=d t$
$\frac{1}{x} d x=\frac{1}{n} d t$
Substituting above equations in $\int \frac{\log x^{n}}{x} d x$ we get,
$\int \frac{1}{n} t d t$
$=\frac{1}{n} \frac{t^{2}}{2}+c$
$=\frac{\left(\log x^{n}\right)^{2}}{2 n}+c$