integral from pi/2 to pi of cos^2(θ)

Lösung

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

\frac{π}{4}

Dezimale

0.78539 \dots

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

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

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

= \frac{1}{2} (π - \frac{π}{2} + 0)

Vereinfache

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

= \frac{π}{4}