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Beliebt Rechnen >

integral of (cos(θ)-cos(2θ))/(1-cos(θ))

Lösung

\int \frac{cos ( θ ) - cos ( 2 θ )}{1 - cos ( θ )} d θ

Lösung

θ + 2 sin (θ) + C

Schritte zur Lösung

\int \frac{cos ( θ ) - cos ( 2 θ )}{1 - cos ( θ )} d θ

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Multipliziere aus

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

Wende die Summenregel an:

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

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

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

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

\int \frac{cos ( θ )}{1 - cos ( θ )} d θ = - cot (\frac{θ}{2}) - θ

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

Vereinfache

- (- cot (\frac{θ}{2}) - sin (θ) - θ) + sin (θ) + θ - cot (\frac{θ}{2}) - θ : θ + 2 sin (θ)

= θ + 2 sin (θ)

Füge eine Konstante zur Lösung hinzu

= θ + 2 sin (θ) + C

Graph

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