(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The question asks that you prove that

[tex]\int\frac{sin^{2}x}{x^2}dx = \pi / 2[/tex]

The integral is from zero to infinity, but I don't know how to add those in latex.

2. Relevant equations

Use a contour integral to get around the pole at z = 0. The problem is, I'm really really foggy on how to do that. Professor says residue theorem doesn't need to be used--and besides, isn't the residue at z = 0 just 0 anyway?

3. The attempt at a solution

Integrate around it by subtracting out a small circular contour around the pole, and then adding it back in.

First, I converted the integral to complex form, and changed the boundaries to -inf to inf, by multiplying by one half. I know this much is right.

[tex]Re \left\{\frac{1}{2} \int\frac{e^{i2z}-1}{z^2}dz\right\}[/tex]

Then I integrate around a small contour: but am I doing this right? Do I just sub these in?

[tex]z = \epsilon e^{i\theta}[/tex] and [tex]dz = i\epsilon e^{i\theta}d\theta[/tex]. Bounds of the integral should be from [tex]\pi[/tex] to 0 if I'm integrating above. Then what? I'm lost. Any integral I try to solve from then on just gives me 0. And what do I do about the exponential in the exponential? I've never seen e to the e before and I don't know what to do with it when integrating, which makes me question whether I'm doing this whole contour thing right. Help!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Integrating around contour (Cauchy)

**Physics Forums | Science Articles, Homework Help, Discussion**