Question

Solve $\int \frac{{\left(\log x\right)}^2}{x}$

Answer 1

Consider the given integral.

$I=\int \frac{{\left(\log x\right)}^2}{x}dx$

Let $t=\log x$

$dt=\frac{dx}{x}$

Therefore,

$I=\int t^{2}dt$

$I=\frac{t^3}{3}+C$

On putting the value of $t$, we get

$I=\frac{{\left(\log x\right)}^3}{3}+C$

Hence, this is the answer.