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

The integral of e^{-iax} is 1/a sin(ax)+i 1/a cos(ax)+C

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integral of e^{-iax}

\int e^{- ia x} d x

\frac{1}{a} sin (a x) + i \frac{1}{a} cos (a x) + C

Solution steps

\int e^{- ia x} d x

Simplify

e^{- ia x} : cos (a x) - i sin (a x)

= \int cos (a x) - i sin (a x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos (a x) d x - \int i sin (a x) d x

\int cos (a x) d x = \frac{1}{a} sin (a x)

\int i sin (a x) d x = - i \frac{1}{a} cos (a x)

= \frac{1}{a} sin (a x) - (- i \frac{1}{a} cos (a x))

Simplify

\frac{1}{a} sin (a x) - (- i \frac{1}{a} cos (a x)) : \frac{1}{a} sin (a x) + i \frac{1}{a} cos (a x)

= \frac{1}{a} sin (a x) + i \frac{1}{a} cos (a x)

Add a constant to the solution

= \frac{1}{a} sin (a x) + i \frac{1}{a} cos (a x) + C

What is the integral of e^{-iax} ?
The integral of e^{-iax} is 1/a sin(ax)+i 1/a cos(ax)+C