Compute $\iint_{\Omega}(x+y)^ndx\,dy$ for $n>0$

Question

Let $n > 0$. Compute the integral

$$\iint_{\Omega}(x+y)^ndx\,dy\,,$$

where $\Omega =\left \{ \left. \right (x,y):x\geq 0,y\geq,x+y\leq 1\} \right.$

Answer 1

Change variables $(x,y)\to (u,v)$ taking $u=x+y$ and $uv=x$.

Then $x>0,y>0,x+y<1\implies0<u,v<1$ and $dx\,dy=u\,du\,dv$.

Hence, $$\iint_{\Omega} (x+y)^n\,dx\,dy=\int_0^1\int_0^1 u^{n+1}\,du\,dv$$