Primitive of Exponential Integral Function

Theorem
Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:


 * $\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$

Then:
 * $\ds \int \map \Ei x \rd x = x \map \Ei x - e^{-x} + C$

Proof
By Derivative of Exponential Integral Function, we have:


 * $\ds \frac \d {\d x} \paren {\map \Ei x} = -\frac {e^{-x} } x$

So: