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Solving limit integral

Given the integral
$$\int_0^\infty \frac{1}{x^2 + a^2}dx,$$
is it possible to find the solution as a function of $a$, where $a \in [0,\infty)$?
The textbook uses complex analysis to find the solution, but I’d prefer to solve the integral using regular techniques.

A:

You may solve this using the substitution $x = \tan \theta$ to get
$$\int_0^\infty \frac{1}{x^2 + a^2}\,dx = \int_0^{\pi/2} \frac{1}{a^2 + \tan^2 \theta}\,d\theta$$
But $\tan^2 \theta = \frac{1 – \cos 2\theta}{\sin^2 \theta}$, so the last integral gives us
$$\int_0^{\pi/2} \frac{1}{a^2 + \frac{1 – \cos 2\theta}{\sin^2 \theta}}\,d\theta$$
Now the integral is very easy to solve:
$$\int \frac{1}{a^2 + \frac{1 – \cos 2\theta}{\sin^2 \theta}}\,d