Here is a more interesting problem to consider. We want to evaluate the improper integral $$\int_0^\infty \frac{\tan^{-1}(6 x) - \tan^{-1}(3 x)}{x}\, dx.$$ Do it by rewriting the numerator of the integrand as $\int_{f(x)}^{g(x)} h(y) dy$ for appropriate $f, g, h$ and then reversing the order of integration in the resulting double integral.

The value of the improper intgeral is .