Definite integral of $\frac{\sin(x)}{x}$ $ $ 
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.

For large $a$, you can do asymptotics and find
$$
\int_{-a}^a \frac{\sin x}{x}dx \sim \pi - \left( \frac2{a} - \frac4{a^3}\right)\cos a  - \left( \frac2{a^2} - \frac{12}{a^4}\right)\sin a  
$$
For $a$ as large as 400, this will be accurate to better than a part in a million.
