integral from 0 to a of sin^2((3pix)/a)

Lösung

\int_{0}^{a} sin^{2} (\frac{3 π x}{a}) d x

\frac{1}{2} a

Schritte zur Lösung

\int_{0}^{a} sin^{2} (\frac{3 π x}{a}) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{a} \frac{1}{2} (1 - cos (\frac{6 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{6 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{6 π x}{a}) d x)

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

\int_{0}^{a} cos (\frac{6 π x}{a}) d x = 0

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

Vereinfache

= \frac{1}{2} a

\int_{- 3}^{3} (x + 6) 9 - x^{2} d x

\int_{0}^{x} e^{- t^{5}} d t

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

\int_{1}^{8} \frac{2}{x ^{\frac{1}{3}}} d x

\int_{0}^{2} \frac{1}{7 x + 10} d x