integral from-pi/2 to pi/2 of cos^2(θ)

Lösung

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

\frac{π}{2}

Dezimale

1.57079 \dots

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{- \frac{π}{2}}^{\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

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

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

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

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

= \frac{1}{2} (π + 0)

Vereinfache

\frac{1}{2} (π + 0) : \frac{π}{2}

= \frac{π}{2}

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