Question

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

Answer 1

By using the substitution `u=lnx`,
`int{(lnx)^2}/xdx={(lnx)^3}/3+C`

Let `u=lnx`.
By taking the derivative with respect to `x`,
`{du}/{dx}=1/x`
By taking the reciprocal,
`{dx}/{du}=x`
By multiplying by `du`,
`dx=xdu`

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=lnx` back in,
`={(lnx)^3}/3+C`