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

Lösung

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

\frac{π ^{2}}{4}

Dezimale

2.46740 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Vereinfache

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= π \cdot \int_{0}^{\frac{π}{2}} \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

= π \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 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

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

\int_{0}^{\frac{π}{2}} 1 d x = \frac{π}{2}

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

= π \frac{1}{2} (\frac{π}{2} - 0)

Vereinfache

π \frac{1}{2} (\frac{π}{2} - 0) : \frac{π ^{2}}{4}

= \frac{π ^{2}}{4}

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