integral from-pi to pi of cos^2(kx)

Lösung

\int_{- π}^{π} cos^{2} (k x) d x

π + \frac{1}{2 k} sin (2 πk)

Schritte zur Lösung

\int_{- π}^{π} cos^{2} (k x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{- π}^{π} \frac{1 + cos ( 2 k x )}{2} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{- π}^{π} 1 + cos (2 k 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

= \frac{1}{2} (\int_{- π}^{π} 1 d x + \int_{- π}^{π} cos (2 k x) d x)

\int_{- π}^{π} 1 d x = 2 π

\int_{- π}^{π} cos (2 k x) d x = \frac{sin ( 2 πk )}{k}

= \frac{1}{2} (2 π + \frac{sin ( 2 πk )}{k})

Vereinfache

\frac{1}{2} (2 π + \frac{sin ( 2 πk )}{k}) : π + \frac{1}{2 k} sin (2 πk)

= π + \frac{1}{2 k} sin (2 πk)

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