# Approximate integral for exp(-x)/x

1. Apr 2, 2013

### Irid

I need to evaluate this integral
$\int_{x_0}^{\infty} \frac{\exp(-x)}{x}\, dx$
for
$x_0 \ll 1$

Is there any cute approximate analytical expression? All I get are diverging series all over the place. The function is obviously finite...

2. Apr 2, 2013

### kamke

Hi,

Prudnikov: Integrals and series

∫(x,∞) e^(-ax)/x dx = -Ei(-ax) a>0

Ei(x) = ∫(-∞,x) e^t/t dt is the exponential integral

Bronstejn

Ei(x) = ∫(-∞,x) e^t/dt = C + ln|x| + x + x*x/(2*2!) + ... + x^n/(n*n!) + ...

C = 0.577215665

kamke

Last edited: Apr 2, 2013