how can I solve this double integral?

Answer 1

[John D'Errico]

1. Why does this look like homework?
2. Why do you assume it has a finite solution?

The integral over y is trivial.

syms y phi
K = (1-y)*exp(y/sin(phi))*cos(phi)^2*sin(phi)*phi

K = 

F = int(K,y,[0,1])

F = 

fplot(F,[0,2*pi])
ylim([-1,5])
yline(0)
grid on

So the integral over phi seems to be not well defined. That part has singularities at 0 and pi. And they will not be well behaved. (The integral will be unbounded.) What does integral tell us?

integral(matlabFunction(F),0,2*pi)

Warning: Inf or NaN value encountered.

ans = Inf

So integral seems to think it is not well posed. We could spend a little time with some simple approximations to understand why not. Or, you can just accept my claim that this kernel is not finitely integrable over that domain.

And that suggests this is not homework. Just not something you can solve. Not all problems you can write down have solutions. In fact, most of the things you just randomly choose to write down will have no solution. Don't ask my why not, as I can offer no solution. ;-)

[Bajdar Nouredine]

Dear @John D'Errico your answer was really helpful, I put max. limit as 2pi instead of pi/2, actually this is not homework , it is part of integration as i found in an article, my aim was to know how integration could be done on two functions one belongs to y and other to theta or phi.