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Beliebt Rechnen >

integral of 1/2 (1+sin(θ))^2

Lösung

\int \frac{1}{2} (1 + sin (θ))^{2} d θ

Lösung

\frac{1}{8} (6 θ - 8 cos (θ) - sin (2 θ)) + C

Schritte zur Lösung

\int \frac{1}{2} (1 + sin (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 (1 + sin (θ))^{2} d θ

Multipliziere aus

(1 + sin (θ))^{2} : 1 + 2 sin (θ) + sin^{2} (θ)

= \frac{1}{2} \cdot \int 1 + 2 sin (θ) + sin^{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 1 d θ + \int 2 sin (θ) d θ + \int sin^{2} (θ) d θ)

\int 1 d θ = θ

\int 2 sin (θ) d θ = - 2 cos (θ)

\int sin^{2} (θ) d θ = \frac{1}{2} (θ - \frac{1}{2} sin (2 θ))

= \frac{1}{2} (θ - 2 cos (θ) + \frac{1}{2} (θ - \frac{1}{2} sin (2 θ)))

Vereinfache

\frac{1}{2} (θ - 2 cos (θ) + \frac{1}{2} (θ - \frac{1}{2} sin (2 θ))) : \frac{1}{8} (6 θ - 8 cos (θ) - sin (2 θ))

= \frac{1}{8} (6 θ - 8 cos (θ) - sin (2 θ))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} (6 θ - 8 cos (θ) - sin (2 θ)) + C

Graph

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