integral from 0 to L of sin^2((2pix)/L)

Lösung

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

\frac{1}{2} L

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{L} \frac{1}{2} (1 - cos (\frac{4 π 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{4 π 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{4 π x}{L}) d x)

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

\int_{0}^{L} cos (\frac{4 π x}{L}) d x = 0

= \frac{1}{2} (L - 0)

Vereinfache

= \frac{1}{2} L

\int_{- 4}^{4} (\frac{x}{2} + 1) d x

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\int_{0}^{π} cos (n) x d x

\int_{2}^{3} (3 x^{2} - \frac{x ^{3}}{4}) d x

\int_{0}^{2 π} - 4 d t