Using integration by parts and the formula $\int {f(x)}\,dx = xf(x)-\int {xf'(x)}\,dx$, one could prove the following:

Fact: If $f$ and $g$ are inverse functions and $f'$ is continuous, then $$\int_{a}^{\,b} {f(x)}\, dx = bf(b) - af(a) - \int_{f(a)}^{\,f(b)} {g(y)}\, dy$$

Use the formula given in the fact to evaluate $\displaystyle \int_{1}^{\,e} {3 \ln(x)}\, dx.$