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I see this integral too much in QFT books when making loop calculations :

## \int_{0}^{1}~ dx~ dy~ dz~ \delta (x+y+z-1) = \int_{0}^{1} dz \int_{0}^{1-z} dy ##

Can any one explain how did we get this ? I mean it's apparent that ##\int_{0}^{1}~ dx \delta (x+y+z-1) ## have been evaluated and equals one, but when applying ##\delta## when integrating ##dy## why it equals ## 1-z## , not ##1-z-x ## ?

Also suppose now I have the following function:

## \int_{0}^{1}~ dx~ dy~ dz~ \log ~\frac{m_1(x+y) + z m_2}{m_1(x+y)+ z(m_2+ m_3)} ##

how can this evaluated ? or even a simpler function:

## \int_{0}^{1}~ dx~ dy~ dz~ \delta (x+y+z-1) ~ (x+y +z) ##

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# I ## \int ~ dx dy dz ~ f(x,y,z)~ \delta (x+y+z-1)##

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