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## \int wdw = \frac{1}{2}w^{2} ##

now if w=x+y,

## \int (x+y)(dx+dw)= \int xdx + \int ydx + \int xdy + \int ydy ##

which can be evaluated and gives

##\int(x+y)(dx+dy)=\frac{1}{2}x^{2}+2xy+\frac{1}{2}y^{2}##

but

##\frac{1}{2}x^{2}+2xy+\frac{1}{2}y^{2} \neq \frac{1}{2}w^{2}##

can someone explain why?

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# Integral inconsistency with variable change

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