Nonelementary integral

[nawidgc]

I have a function namely cos(x)/x^2 which I need to integrate in the
limits of x = -1 to x = +1.

If we plot the integrand, we can see that the integrand is positive all the time inside the limits of integration. Also note that the integrand is singular in the given limit at x=0.

Now since this integral is not a simple one to handle, I resorted to
Mathematica for solving it. Strangely, Mathematica returns a value of
negative 2.97 for the integral (I only remember the first two digits
after decimal point). Question is, when the integrand is positive all
the time, how can the integral be negative. When I try some online
integration tools for this function, they return with message that
this integral is likely to be a nonelementary kind. I know funny
things do happen at the singularities but the answer from Mathematica
does not make sense to me. Does anyone have any comments on this?
Thanks for your help.

gcd.

PS. I tried to evaluate the integral with Trapezoid rule and the answer returned does make sense. It is 0.4782921717 for 82 sampling points.

 

[galactus]

Are you sure you typed it into the Integrator correctly?. I just did as a check and it said, "integral does not converge". As it should have. If you graph it, notice the vertical asymptote at x=0.

This is a divergent integral.

 

[Deleted member 4993]

I got "does not converge" statement from Wolfram-α also.

 

[galactus]

You could consider a comparison test.



Since \(\displaystyle \int_{-1}^{1}\frac{1}{x^{2}}dx\) diverges, so does your integral.

 

[Deleted member 4993]

You could consider a comparison test.



Since \(\displaystyle \int_{-1}^{1}\frac{1}{x^{2}}dx\) diverges, so does your integral.

But we have:

\(\displaystyle \dfrac{cos(x)}{x^2} \le \dfrac{1}{x^2}\) → divergence is not guaranteed.