I was given a simple problem on uniform convergence of parametric integrals. The integral was $$ I(a)=\int_0^{+\infty}\frac{\cos(x)}{x+a}dx, \qquad a>0 $$ It seems like a straightforward application of the Abel-Dirichlet test. Just to be clear here is my argument. Set $f(x,a)=\cos(x)$, $g(x,a)=\f...