TITLE: Definite integral of $\frac{\sin(x)}{x}$
QUESTION [11 upvotes]: $ $ 
I'm wondering how to approach the following definite integral:
$$\int_{-400}^{400} \frac{\sin(x)}{x}dx = \int_{-400}^{400} \DeclareMathOperator{\sinc}{sinc} \sinc(x) dx$$
I tried taylor expanding and integrating the polynomial but then I get a divergent series.

REPLY [3 votes]: $$\int_{-a}^{a}\frac{\sin(x)}x\,dx = \pi - 2\int_{a}^{\infty}\frac{\sin(x)}x\,dx$$
Integrating by parts the RHS integral:
$$
\int_{a}^{\infty}\frac{\sin(x)}x\,dx = \frac{\cos a}a - \int_{a}^{\infty}\frac{\cos(x)}{x^2}\,dx
$$
and you have the trivial bounding
$$
\left|\int_{a}^{\infty}\frac{\cos(x)}{x^2}\,dx\right|\le\frac1a.
$$
Repeating the procedure you can get better bounds.