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.

Solution 1:

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.

Solution 2:

$$\int_{-a}^{a}\frac{\sin(x)}x\,dx = \pi - 2\int_{a}^{\infty}\frac{\sin(x)}x\,dx$$ 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.