Q.Evaluate : $\displaystyle\int_{b}^{\infty}\dfrac{1}{a^2+x^2}\,dx$, $a>0$, $b\in\mathbb{R}$.

A Riemann (definite) integral $\int_a^b f(x)\,dx$ requires a finite interval $[a,b]$ with $f$ finite throughout. An improper integral of the first kind relaxes the finiteness of the interval — one or both limits of integration are $\pm\infty$ — and is defined as a limit of ordinary Riemann integrals:

If the limit exists (and is finite), the improper integral is said to converge; otherwise it diverges. By the Second Fundamental Theorem, if $F$ is an anti-derivative of $f$, then $\int_a^\infty f(x)\,dx=\lim_{t\to\infty}\big[F(t)-F(a)\big]$ — so in practice one finds $F$, substitutes $t$ for the infinite endpoint, and takes the limit termwise.

Recurring limiting facts used throughout this chapter:

• $\displaystyle\lim_{t\to\infty}\tan^{-1}t=\dfrac{\pi}{2}$, so $\int_0^\infty \dfrac{dx}{a^2+x^2}=\dfrac{\pi}{2a}$ for $a>0$. …