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Beliebt Rechnen >

integral from 0 to l of sin^2((npix)/l)

Lösung

\int_{0}^{l} sin^{2} (\frac{nπ x}{l}) d x

Lösung

\frac{1}{2} (l - \frac{l sin ( 2 πn )}{2 πn})

Schritte zur Lösung

\int_{0}^{l} sin^{2} (\frac{nπ x}{l}) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{l} \frac{1}{2} (1 - cos (\frac{2 πn x}{l})) 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}^{l} 1 - cos (\frac{2 πn x}{l}) 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}^{l} 1 d x - \int_{0}^{l} cos (\frac{2 πn x}{l}) d x)

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

\int_{0}^{l} cos (\frac{2 πn x}{l}) d x = \frac{l sin ( 2 πn )}{2 πn}

= \frac{1}{2} (l - \frac{l sin ( 2 πn )}{2 πn})

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