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

Lösung

\int_{0}^{π} sin^{2} (y) d y

\frac{π}{2}

Dezimale

1.57079 \dots

Schritte zur Lösung

\int_{0}^{π} sin^{2} (y) d y

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Entferne die Konstante:

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

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

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

\int_{0}^{π} 1 d y = π

\int_{0}^{π} cos (2 y) d y = 0

= \frac{1}{2} (π - 0)

Vereinfache

\frac{1}{2} (π - 0) : \frac{π}{2}

= \frac{π}{2}

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