What is the integral of e^{-pix} ?

The integral of e^{-pix} is -1/pi e^{-pix}+C

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integral of e^{-pix}

\int e^{- π x} d x

- \frac{1}{π} e^{- π x} + C

Solution steps

\int e^{- π x} d x

Apply Integration By Parts

: e^{- π x} x - \int - π e^{- π x} x d x

= e^{- π x} x - \int - π e^{- π x} x d x

\int - π e^{- π x} x d x = e^{- π x} x + \frac{1}{π} e^{- π x}

= e^{- π x} x - (e^{- π x} x + \frac{1}{π} e^{- π x})

Simplify

= - \frac{1}{π} e^{- π x}

Add a constant to the solution

= - \frac{1}{π} e^{- π x} + C

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