- #1

- 1,456

- 44

## Main Question or Discussion Point

I have the integral ##\displaystyle \int_{- \infty}^{\infty} \frac{\cos x}{x^2+1} dx##. We are going to use differentiation under the integral sign, so we let ##\displaystyle I(t) = \int_{- \infty}^{\infty} \frac{\cos tx}{x^2+1} dx##, and then, after manipulation, ##\displaystyle I'(t) = \int_{- \infty}^{\infty} \frac{\sin tx}{x(x^2+1)} dx - \int_{- \infty}^{\infty} \frac{\sin tx}{x} dx##. My question lies in the rightmost integral. In a solution I've seen, the rightmost integral is linked to the Dirichlet integral: https://en.wikipedia.org/wiki/Dirichlet_integral. And so ##\pi## is simply substituted for this expression. What I don't understand is how can it be linked to this known integral when there is that ##t## in the argument of ##\sin##?