integral from 0 to a of sin^2(kx)

Lösung

\int_{0}^{a} sin^{2} (k x) d x

\frac{1}{2} (a - \frac{sin ( 2 ak )}{2 k})

Schritte zur Lösung

\int_{0}^{a} sin^{2} (k x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{a} \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_{0}^{a} 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_{0}^{a} 1 d x - \int_{0}^{a} cos (2 k x) d x)

\int_{0}^{a} 1 d x = a

\int_{0}^{a} cos (2 k x) d x = \frac{sin ( 2 ak )}{2 k}

= \frac{1}{2} (a - \frac{sin ( 2 ak )}{2 k})

\int_{1}^{2} 2 x^{2} + x d x

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