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

Lösung

\int_{0}^{2 π} (sin (x))^{2} d x

π

Dezimale

3.14159 \dots

Schritte zur Lösung

\int_{0}^{2 π} (sin (x))^{2} d x

Vereinfache

= \int_{0}^{2 π} sin^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{2 π} \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

= \frac{1}{2} \cdot \int_{0}^{2 π} 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

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

\int_{0}^{2 π} 1 d x = 2 π

\int_{0}^{2 π} cos (2 x) d x = 0

= \frac{1}{2} (2 π - 0)

Vereinfache

\frac{1}{2} (2 π - 0) : π

= π

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{3} (x) d x

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

\int_{- 1}^{1} x cos (x) d x

\int_{0}^{π} x^{2} d x

\int_{0}^{π} cos^{2} (θ) sin (θ) d θ