Calculating integral / Changing order of integration $\int_0^1(\int_1^2(\int_1^2 z^{x+y}dx)dy)dz)$

Question

$$\int_0^1(\int_1^2(\int_1^2 z^{x+y}dx)dy)dz)$$

How can I calculate this integral? And when am I allowed to change the order of integration?

I thought about calculating $\int z^{x+y} dz=\frac{z^{x+y+1}}{x+y+1}$ first, but I am looking for a justification to integrate this term first and what to do next.

Answer 1

$I = \int_0^1(\int_1^2(\int_1^2 z^{x+y}dx)dy)dz)$

$I =\int_1^2\int_1^2\int_0^1z^{x+y}\,dz\,dx\,dy$

$I= \int_1^2\int_1^2\frac{z^{x+y+1}}{x+y+1}\bigg|_0^1\,dx\,dy$

$I =\int_1^2\int_1^2\frac{1}{x+y+1}\,dx\,dy$

$I=\int_1^2\ln(x+y+1)\bigg|_1^2\,dy$

$I = \int_1^2(\ln(y+3)-\ln(y+2))\,dy$

$I =\int_1^2\ln(y+3)\,dy-\int_1^2\ln(y+2)\,dy$

$I = (y+3)\big[\ln(y+3)-1\big]_1^2-(y+2)\big[\ln(y+2)-1\big]_1^2$

$I = 5[\ln(5)-1]-(4)[\ln(4)-1]-(4)[\ln(4)-1]+3[\ln(3)-1] $

$I =5\ln(5)-8\ln(4)+3\ln(3)$