Solution
= Solution
Use <integration by parts> with $u=(\ln x)^{n+1}$ and $dv=dx$. Since $du=(n+1)(\ln x)^n\,dx/x$, this yields
$$
\boxed{I_{n+1}=x(\ln x)^{n+1}-(n+1)I_n}.
$$
As an identity between <indefinite integrals>, the formula is understood up to an additive constant.