Integration of some function containing a singularity

Dear All,

These days I am working on some integration integrate(cos(x)*sin(x)/x^t),x(0,a),a is a value. the value fot t is 1, 2, 3. The function also have the form cosx*cosx and sinx*sinx. The value x = 0 is the singularity. How to overcome this singularity. The function quadgk can integrate some funtion with endpoint singularity. x = 0 is the end point singularity. But the integral is not correct when use quadgk. How to solve this problem.

Thank you for your concern.

Answer 1

[Mike Hosea]

QUADGK has no trouble when t=1, but for t=2,3,4,... the correct answer, if I am not mistaken, is Inf. You were expecting QUADGK to return Inf for you?

[Walter Roberson]

As a point of interest, could someone run

syms x

syms t positive

int(cos(x)*sin(x)/x^t,x=0..pi)

through MuPAD ?

Maple mis-integrates in this case (which I just reported a moment ago.)

[Xianjie]

But when the x=0 is not selected, choose x=0.001, the integration is ok. How about this phenomenon. Is there some numerical evaluate the results? I calculate the strain energy of circular plate. x=0 is the center of circular plate.

Thanks

[Mike Hosea]

I believe QUADGK handles that case properly, though you may well run into precision issues.