# Integration technique of writing $\int_0^{\infty}$ as $\int_0^1 + \int_1^{\infty}$ and using the substitution $\frac{1}{x} \leftrightarrow u$

I've recently seen this idea applied a few times to evaluate otherwise difficult integrals. Suppose you have an integral $\int_0^{\infty} f(x)\,\mathrm{d}x$. You can break up the domain of integration and make the substitution $\frac{1}{x}\leftrightarrow u$ to arrive at an integral which may be easier to evaluate.

\begin{align} \int\limits_0^{\infty} f(x)\,\mathrm{d}x \;\;&=\;\; \int\limits_0^1 f(x)\,\mathrm{d}x + \int\limits_1^{\infty} f(u)\,\mathrm{d}u \\\;\;&=\;\;\int\limits_0^1 f(x)\,\mathrm{d}x + \int\limits_0^1 \frac{f\!\left(\frac{1}{x}\right)}{x^2}\,\mathrm{d}x \;\;=\;\; \int\limits_0^1 \frac{x^2f(x)+f\!\left(\frac{1}{x}\right)}{x^2}\,\mathrm{d}x \end{align}

Does this technique have a name? Are there broad families of functions for which this trick is essential for integration, or does this only help in very specific cases? Can this idea be applied generally enough to be worth showing students?

• A neat generalisation of the first is $\int_{\mathbb{R}} f(x)\,\mathrm{d}x=\int_{\mathbb{R}} f\left(x-\sum_{i=1}^n \frac{\alpha_i}{x+\beta_i}\right)\,\mathrm{d}x$ for any $\alpha_i \in\mathbb{R}^+$ and $\beta_i\in\mathbb{R}$ – jcg Jul 28 '19 at 0:25