# How do you use Integration by Substitution to find int(ln(x))^2/xdx?

Sep 6, 2014

By using the substitution $u = \ln x$,
$\int \frac{{\left(\ln x\right)}^{2}}{x} \mathrm{dx} = \frac{{\left(\ln x\right)}^{3}}{3} + C$

Let $u = \ln x$.
By taking the derivative with respect to $x$,
$\frac{\mathrm{du}}{\mathrm{dx}} = \frac{1}{x}$
By taking the reciprocal,
$\frac{\mathrm{dx}}{\mathrm{du}} = x$
By multiplying by $\mathrm{du}$,
$\mathrm{dx} = x \mathrm{du}$

Now, we can rewrite the integral in terms of $u$,
int{(lnx)^2}/xdx =intu^2/x xdu=int u^2 du
by Power Rule,
$= {u}^{3} / 3 + C$
by putting $u = \ln x$ back in,
$= \frac{{\left(\ln x\right)}^{3}}{3} + C$