3.4: Change of Variables
Another useful technique for solving integrals is to change variables. Consider the integral \[\int_0^\infty \frac{dx}{x^2 + 1}.\] We can solve this by making a change of variables \(x = \tan(u)\) . This involves (i) replacing all occurrences of \(x\) in the integrand with \(\tan(u)\) , (ii) replacing the integral limits, and (iii) replacing \(dx\) with \((dx/du) \, du = 1/[\cos(u)]^2 du\) : \[\begin{align} \int_0^\infty \frac{dx}{x^2 + 1} &= \int_0^{\pi/2} \frac{1}{[\tan(u)]^2 + 1} \cdot \frac{1}{[\cos(u)]^2} \; du \\ &= \int_0^{\pi/2} \frac{1}{[\sin(u)]^2 + [\cos(u)]^2} \; du.\end{align}\] Due to the Pythagorean theorem, the integrand reduces to 1, so \[\int_0^\infty \frac{dx}{x^2 + 1} = \int_0^{\pi/2} du = \frac{\pi}{2}.\] Clearly, this technique often requires some cleverness and/or trial-and-error in choosing the right change of variables.