integral from 0 to 2pi of (sin^2(t))

Lösung

\int_{0}^{2 π} (sin^{2} (t)) d t

π

Dezimale

3.14159 \dots

Schritte zur Lösung

\int_{0}^{2 π} sin^{2} (t) d t

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{2 π} 1 - cos (2 t) d t

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}^{2 π} 1 d t - \int_{0}^{2 π} cos (2 t) d t)

\int_{0}^{2 π} 1 d t = 2 π

\int_{0}^{2 π} cos (2 t) d t = 0

= \frac{1}{2} (2 π - 0)

Vereinfache

\frac{1}{2} (2 π - 0) : π

= π

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