Find the power series of the integrand g(t) about t=0 first. Use
et = 1+t+t2/2!+t3/3!+... =∑n=0∞ tn/n!
et-1 = t+t2/2!+t3/3!+... =∑n=1∞ tn/n!
(et-1)/t = 1+t/2!+t2/3!+... =∑n=1∞ tn-1/n!
Now integrate this series term by term:
∫0x (et-1)/t dt= ∫0x (1+t/2!+t2/3!+...) dt = ∫0x ∑n=1∞ tn-1/n! dt
= x + x2/(2*2!)+x3/(3*3!)+... = ∑n=1∞ xn/(n*n!)
By the ratio test, the series converges everywhere.