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Beliebt Rechnen >

integral from 0 to pi/4 of 25sin^2(x)

Lösung

\int_{0}^{\frac{π}{4}} 25 sin^{2} (x) d x

Lösung

\frac{25 ( π - 2 )}{8}

+1

Dezimale

3.56747 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{4}} 25 sin^{2} (x) d x

Entferne die Konstante:

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

= 25 \cdot \int_{0}^{\frac{π}{4}} sin^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 25 \cdot \int_{0}^{\frac{π}{4}} \frac{1 - cos ( 2 x )}{2} d x

Entferne die Konstante:

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

= 25 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{4}} 1 - cos (2 x) 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

= 25 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{4}} 1 d x - \int_{0}^{\frac{π}{4}} cos (2 x) d x)

\int_{0}^{\frac{π}{4}} 1 d x = \frac{π}{4}

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

= 25 \cdot \frac{1}{2} (\frac{π}{4} - \frac{1}{2})

Vereinfache

25 \cdot \frac{1}{2} (\frac{π}{4} - \frac{1}{2}) : \frac{25 ( π - 2 )}{8}

= \frac{25 ( π - 2 )}{8}

Graph

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