integral from 0 to pi/2 of 8cos^2(θ)

Lösung

\int_{0}^{\frac{π}{2}} 8 cos^{2} (θ) d θ

2 π

Dezimale

6.28318 \dots

Schritte zur Lösung

\int_{0}^{\frac{π}{2}} 8 cos^{2} (θ) d θ

Entferne die Konstante:

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

= 8 \cdot \int_{0}^{\frac{π}{2}} cos^{2} (θ) d θ

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 8 \cdot \int_{0}^{\frac{π}{2}} \frac{1 + cos ( 2 θ )}{2} d θ

Entferne die Konstante:

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

= 8 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 1 + cos (2 θ) d θ

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 8 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d θ + \int_{0}^{\frac{π}{2}} cos (2 θ) d θ)

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

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

= 8 \cdot \frac{1}{2} (\frac{π}{2} + 0)

Vereinfache

8 \cdot \frac{1}{2} (\frac{π}{2} + 0) : 2 π

= 2 π

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