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

Lösung

\int_{- π}^{π} x e^{- in x} d x

\frac{2 πi ( - 1 ) ^{n}}{n}

Schritte zur Lösung

\int_{- π}^{π} x e^{- in x} d x

Vereinfache

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

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

Wende die Summenregel an:

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

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

\int_{- π}^{π} x cos (n x) d x = 0

\int_{- π}^{π} i x sin (n x) d x = - \frac{2 πi ( - 1 ) ^{n}}{n}

= 0 - (- \frac{2 πi ( - 1 ) ^{n}}{n})

Vereinfache

= \frac{2 πi ( - 1 ) ^{n}}{n}

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