Does the Integral of Cos(x)/x² from Negative to Positive Infinity Converge?

[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 don't know what I am talking about), and it gave me the answer -cos(x)/x - SinIntegral(x). SinIntegral is finite at infinity, and 0 at 0, so doesn't this mean it diverges? does the fact that I am looking for the principal part change this?

 

[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 doesn't 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 there's no problem.

 

[M98Ranger]

The principal part of an integral refers to the part of the integral that contributes to its value as the limits of integration approach a singularity. In this case, the singularity is at x=0 where the function cos(x)/x^2 is undefined. The integral may converge or diverge depending on the behavior of the function near this singularity.

In order to calculate the principal part of this integral, you would need to use techniques such as integration by parts or substitution to transform the integral into a form that is easier to evaluate. However, in this case, the integral does not have a closed form solution and therefore, it cannot be evaluated analytically.

As for the result you obtained from the integrator tool, it is important to note that the tool may not always give accurate results for indefinite integrals. In this case, the result you obtained is not the principal part of the integral, but rather the indefinite integral which includes the constant of integration. The constant of integration is necessary to account for the singularity at x=0 and it does not affect the convergence or divergence of the integral.

In conclusion, the integral \int_{-\infty}^{\infty} \frac{cos(x)}{x^2} dx does not have a well-defined principal part and therefore, it cannot be evaluated using traditional methods. It is a type of improper integral that requires advanced techniques such as complex analysis to determine its convergence or divergence.