I am trying to understand what is wrong with the proof posted here that $0=1$ ([source](http://www-math.mit.edu/~tchow/mathstuff/proof.pdf)): 

> 
> Given any $x$, we have (by using the substitution $u=x^2/y$) $$\large\int\_0^1\frac{x^3}{y^2}e^{-x^2/y}\,dy=\Biggl[xe^{-x^2/y}\Biggr]\_0^1=xe^{-x^2}.$$
>  Therefore, for all $x$,
>  \begin{align}
> e^{-x^2}(1-2x^2)&=\frac{d}{dx}(xe^{-x^2})\\[0.5em]
> &= \frac{d}{dx}\int\_0^1\frac{x^3}{y^2}e^{-x^2/y}\,dy\\[0.5em]
> &= \int\_0^1\frac{\partial}{\partial x}\Biggl(\frac{x^3}{y^2}e^{-x^2/y}\Biggr)\,dy\\[0.5em]
> &= \int\_0^1 e^{-x^2/y}\Biggl(\frac{3x^2}{y^2}-\frac{2x^4}{y^3}\Biggr)\,dy.
> \end{align}
>  Now set $x=0$; the left-hand side is $e^0(1-0)=1$, but the right-hand side is $\int\_0^1 0\,dy=0$. 
> 
> 
> 

Surely it must be incorrect somewhere however I am not sure at which step it fails.