Please help with evaluating an integral using the Residue Theorem

Asked Apr 25 '13 at 04:05

Active Apr 25 '13 at 04:13

Viewed 142 times

Use the Residue Theorem to evaluate $\displaystyle\int_{0}^{∞} \frac{\sin^2(x)}{x^2} \, dx$.

Using the trig identity, this is how far I've gotten: let $F(z)=\dfrac{1}{2}\dfrac{1-(e^{2iz})}{z^2}$, and then $\int_{0}^{∞} F(z) \,dz$.

But I'm not sure how to proceed from here. please help!

edited Apr 25 '13 at 04:13

Michael Hardy

asked Apr 25 '13 at 04:05

Kelly

Dear Kelly, One comment is that in these kinds of questions, it is often helpful to replace the integral over the half-line to the integral over the full line. In your case this is easy, because the integrand is even, so the integral you want can be rewritten as $1/2$ times the integral from $-\infty$ to $\infty$. Regards, – Matt E Apr 25 '13 at 04:24
Here is the same problem. – Mhenni Benghorbal Apr 25 '13 at 05:09
You can find the method you want here. – Mhenni Benghorbal Apr 25 '13 at 05:15

0 Answers0