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

Lösung

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

\frac{1}{2} a

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

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

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

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

Vereinfache

= \frac{1}{2} a

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

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

\int_{0}^{1} \frac{1}{( 1 + e ^{x} ) ^{2}} d x

\int_{- 4}^{4} x d x

\int_{0}^{3} (1 - e^{- x}) d x