Question

$\int dx \frac{1+z^x}{1-_{9}x}$

Answer 1

$\int\frac{1+z^x}{\,1-9x}\,dx=-\frac{1}{9}\ln|1-9x|+\frac{z^{1/9}}{9}\,Ei\Bigl(-\frac{\ln z}{9}(1-9x)\Bigr)+C.$

Answer 2

1. Split the integral into two parts.

2. In the first, substitute $u=1-9x$ to integrate $\int\frac{dx}{1-9x}$.

3. In the second, write $z^x=e^{x\ln z}$ and use the same substitution; recognize the definition of the exponential integral $Ei(-Au)$.

4. Combine the answers.