Consider trying to find the indefinite integral of $${1\over e^x+1}.$$
One way to go is to change the variable of integration. This can be done in two ways, viz.: (1) Take $u=e^x+1,$ which transforms the integral into $\int du/u(u-1).$ (2) On the other hand, if one takes $v=e^x,$ the integral transforms into $\int dv/v(v+1).$
This is apparently a contradiction. Could someone please show me where the problem is?