Fubini's Theorem to justify Integrability Examples 
I've recently been learning about Fubini and Tonelli's Thms but have no idea how to even approach these questions. 
Any insight and help would be greatly appreciated. 
Thank you! 
 A: This Fubini and Tonelli thing is only about being able to exchange the integrals, given that any of the possibilities (integrating first one, the other, or both together) of the modulus of the integrand can be bounded.
So under this topic, you MUST exchange integrals if you see them.
1) Develop J and exchange the integrals. since there are no singular terms, you don't care about the integral over $[0 , 2\pi]$ as it's over a compact set. The integral with respect to $x$ can be bounded because $| \cos(\cdot ) | \leq 1$, and thus
$$ \int_0^{\pi/2} \int_0^\infty \frac{2}{\pi} | \cos(x \cos(\theta)) | e^{-a x} dx d\theta \leq \int_0^\infty e^{-a x} dx = \frac{1}{a} < \infty $$
This justifies the exchange of integrals and then you go forward. To solve the integral over $x$ I suggest you to integrate by parts twice, you should get something like $-a/(a^2+cos(\theta)^2)$. Then you integrate with respect to theta,  since you know already the solution you will come up with something.
2) As I said, since the topic is Tonelli- Fubini, we must exchange the integrals... there are no? make them appear ! Inside the integral there is a subtraction between terms of the form $f(a)-f(b)$, so we can write this as $\int_b^a f'(y) dy$, when doing this the integral will strongly simplificate, now you will be able proof its boundedness following a similar process as in 1) and then compute it.
