I am currently studying this problem: 14 b)
There you see an integral $$A(r) = \int f(\theta) (-\sin(\phi), \cos(\phi),0) d \Omega$$ where $f$ is the function containing all the rest of the integrand that you see there and I don't see why this integral is not zero? (I am especially referring to the integral in the second row of part b) . I mean clearly: $$\int_{0}^{2\pi} (-\sin(\phi), \cos(\phi),0) d\phi = 0$$ so why does this integral not vanish completely?
Also I don't get why there is this $\phi'$ in the denominator? I mean, don't we have $$||r-r'||= \sqrt{r^2+r'^2-2rr'\cos(\theta-\theta')}$$ so this should not depend on $\phi'$? (This is the reason why I said that $f$ only depends on $\theta$.)