Is this substitution valid?

Question

So I made this identity for Euler's constant $\gamma$:$$\int_{-\infty}^\infty e^{-(e^t+t)}tdt=-\gamma$$But this isn't right for some reason. So what's wrong with my proof?: $$\int_0^\infty e^{-t}\ln tdt=-\gamma$$Make a substitution of $t=\ln u$ to get: $$\int_{-\infty}^\infty \frac{udu}{e^{e^u}e^u}$$The derivative of $\ln t$ is $\frac{1}{t}$ but our substitution turns it to $\frac{1}{e^t}$. Shouldn't $e^{-t}$ map to $e^{-e^t}$? Where is my mistake?