# Principal part of an integral

1. Oct 30, 2004

### StatusX

how would i calculate the principle part of:

$$\int_{-\infty}^{\infty} \frac{cos(x)}{x^2} dx$$

it seems like this would diverge at 0, which might be ok except that it also osciallates and goes to 0 at infinity, so it doesn't look like you could balance the infinity at 0 by extending the bounds to infinity. does this converge, and if so, how would i find the value?

Also, I put this into the Integrator thing(search for integrator if you dont know what im talking about), and it gave me the answer -cos(x)/x - SinIntegral(x). SinIntegral is finite at infinity, and 0 at 0, so doesnt this mean it diverges? does the fact that im looking for the principal part change this?

Last edited: Oct 30, 2004
2. Oct 30, 2004

### StatusX

never mind, it diverges. this was from an assignment, and the question was to evaluate the integral using contours in the complex plane. every example we had done had come out to a finite value, and it doesnt seem to make sense to evaluate a definite integral to get an value of infinity; the integral just diverges, it has no value. but i asked the teacher and he says he was looking for infinity as the answer. anyway, i assumed i was doing something wrong, but now theres no problem.