# How do you integrate int e^x tan x dx  using integration by parts?

$\setminus \tan \left(x\right) = - i \setminus \frac{1 - {e}^{- 2 i x}}{1 + {e}^{- 2 i x}} = - i - 2 i \setminus {\sum}_{k = 1}^{\setminus} \infty {\left(- 1\right)}^{k} {e}^{- 2 i k x}$
 \int e^x \tan(x)\ dx = -i e^x - 2 i \sum_{k=1}^\infty (-1)^k \int e^{(1-2ik)x}\ dx
$= - i {e}^{x} - 2 i \setminus {\sum}_{k = 1}^{\setminus} \infty \setminus \frac{{\left(- 1\right)}^{k}}{1 - 2 i k} {e}^{\left(1 - 2 i k\right) x} + C$