Improper Integrals

The Riemann integral $\int_a^b f(x)\,dx$ (§9.2.1) requires the interval $[a,b]$ to be finite and $f(x)$ to be finite at every point of $[a,b]$. Many physical applications instead need integrals of the shape

where $a$ is real and $f$ is continuous on the interval of integration. These are called improper integrals of the first kind, and are defined as limits of ordinary (proper) Riemann integrals:

If the relevant limit exists (is finite), the improper integral is said to converge; otherwise it diverges.

Practical evaluation. By the Second Fundamental Theorem, there is a function $F(t)$ with $\int_a^t f(x)\,dx=F(t)-F(a)$, so $\displaystyle\int_a^\infty f(x)\,dx=\lim_{t\to\infty}\big[F(t)-F(a)\big]$ — find the ordinary anti-derivative, substitute $t$ for the infinite endpoint, and take the limit afterward.

Worked pattern (Example 9.35). $\displaystyle\int_0^\infty\dfrac{1}{a^2+x^2}\,dx=\left[\dfrac1a\tan^{-1}\dfrac xa\right]_0^\infty=\dfrac1a\left(\dfrac\pi2-0\right)=\dfrac{\pi}{2a}$ (using $\lim_{x\to\infty}\tan^{-1}x=\frac\pi2$); and, since $\frac1{a^2+x^2}$ is even, $\displaystyle\int_{-\infty}^\infty\dfrac{dx}{a^2+x^2}=2\int_0^\infty\dfrac{dx}{a^2+x^2}=\dfrac{\pi}{a}$. …