Question

Solve :
$\int \frac{\log x^3}{x}dx$

Answer 1

We have,

$I=\int \frac{\log x^3}{x}dx$

Let

$t=\log x^3$

$\frac{dt}{dx}=\frac{1}{x^3}\times 3x^2$

$\frac{dt}{dx}=\frac{3}{x}$

$\frac{dt}{3}=\frac{dx}{x}$

Therefore,

$I=\frac{1}{3}\int tdt$

$I=\frac{1}{3}\frac{t^2}{2}+C$

$I=\frac{t^2}{6}+C$

On putting the value of $t$, we get

$I=\frac{(\log x^3{)}^2}{6}+C$

Hence, this is the answer.