integral from 0 to L of sin^2(ax)

Lösung

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

\frac{1}{2} L - \frac{1}{4 a} sin (2 a L)

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

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

\int_{0}^{L} cos^{2} (a x) d x = \frac{1}{2} (L + \frac{sin ( 2 a L )}{2 a})

= L - \frac{1}{2} (L + \frac{sin ( 2 a L )}{2 a})

Vereinfache

L - \frac{1}{2} (L + \frac{sin ( 2 a L )}{2 a}) : \frac{1}{2} L - \frac{1}{4 a} sin (2 a L)

= \frac{1}{2} L - \frac{1}{4 a} sin (2 a L)

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