My-old-problem-e-tan-x-dx-

Question Number 186192 by normans last updated on 02/Feb/23

\boldsymbol M y \boldsymbol o l d \boldsymbol p r o b l e m

\int \boldsymbol e^{\boldsymbol t a n \boldsymbol x} \boldsymbol d x

Answered by MJS_new last updated on 02/Feb/23

\int e^{\tan x} d x =

[t = \tan x \to d x = \frac{d t}{t^{2} + 1}]

= \int \frac{e^{t}}{t^{2} + 1} d t = \int \frac{e^{t}}{(t - i) (t + i)} d t =

= \frac{i}{2} \int (\frac{e^{t}}{t + i} - \frac{e^{t}}{t - i}) d t =

= \frac{i}{2} (e^{- i} Ei (t + i) - e^{i} Ei (t - 1)) =

\dots

for the Integral Exponential Function Ei (x)

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