What is the integral from-pi to pi of e^{-inx} ?

The integral from-pi to pi of e^{-inx} is 0

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integral from-pi to pi of e^{-inx}

\int_{- π}^{π} e^{- in x} d x

0

Solution steps

\int_{- π}^{π} e^{- in x} d x

Simplify

e^{- in x} : cos (n x) - i sin (n x)

= \int_{- π}^{π} cos (n x) - i sin (n 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_{- π}^{π} cos (n x) d x - \int_{- π}^{π} i sin (n x) d x

\int_{- π}^{π} cos (n x) d x = 0

\int_{- π}^{π} i sin (n x) d x = 0

= 0 - 0

Simplify

= 0

What is the integral from-pi to pi of e^{-inx} ?
The integral from-pi to pi of e^{-inx} is 0