Integral of xcos(xy) over a rectangle

[carlosbgois]

Hi there. I'm having some trouble with some double integrals here. All of them are to be evaluated on the rectangle$_{}$ $1≤x≤2, 0≤y≤1$, and the functions are:

1: $\int\int_{A}\frac{1}{x+y}dxdy$.
On this one I made $α(y)=\int^{2}_{1}\frac{1}{x+y}dx=ln(\frac{2+y}{1+y})$, and finally I should evaluate $\int^{1}_{0}α(y)dy$, and this is where I got stuck.

2: $\int\int_{A}xcos(xy)dxdy$.
I again have evaluated $α(y)$, using integration by parts, but then I got stuck with some integrals like $\int\frac{sin(y)}{y}dy$, $\int\frac{cos(2y)}{y^{2}}dy$, and so on, which I couldn't find the primitive.

Any tips on how to evaluate those?

 

[arildno]

You are fully allowed, by Fubini's theorem, to switch the order of integration.

 

[carlosbgois]

That's it, for both cases. Many thanks.