I need to calculate the value of these integrals:
$$\int\int_{(0,e)\times(0,+\infty)}\frac{x\log x}{1+(xy)^2}dxdy,$$ $$\int\int_{(0,+\infty)\times(0,+\infty)}\frac{x \arctan x}{(1+x^2)(x^2+y^2)}dxdy.$$
I've tried using Tonelli's theorem, but I'm not able to calculate the value of the expressions that I get.
In addition, I would appreciate it if you could recommend me a book with solved problems on multivariable integration.
Thanks in advance!
In both cases, you may integrate $y$ first to get $$ I_1 = \int_0^{e}\frac{\pi \log (x)}{2}dx $$ and $$ I_2=\int_0^{\infty}\frac{\pi \tan ^{-1}(x)}{2 \left(x^2+1\right)}dx. $$ Both of these integrals are easily computed via $$ \int \log(x) dx = x(\log(x)-1) + C $$ and $$ \int \frac{\tan ^{-1}(x)}{x^2+1} = \frac{1}{2} \tan ^{-1}(x)^2 + C $$ to give $$ I_1=0 $$ and $$ I_2 = \frac{\pi ^3}{16}. $$