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Beliebt Rechnen >

integral of (1-cos(θ))^3

Lösung

\int (1 - cos (θ))^{3} d θ

Lösung

\frac{5}{2} θ + \frac{3}{4} sin (2 θ) - 4 sin (θ) + \frac{1}{3} sin^{3} (θ) + C

Schritte zur Lösung

\int (1 - cos (θ))^{3} d θ

Multipliziere aus

(1 - cos (θ))^{3} : 1 - 3 cos (θ) + 3 cos^{2} (θ) - cos^{3} (θ)

= \int 1 - 3 cos (θ) + 3 cos^{2} (θ) - cos^{3} (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

= \int 1 d θ - \int 3 cos (θ) d θ + \int 3 cos^{2} (θ) d θ - \int cos^{3} (θ) d θ

\int 1 d θ = θ

\int 3 cos (θ) d θ = 3 sin (θ)

\int 3 cos^{2} (θ) d θ = \frac{3}{2} (θ + \frac{1}{2} sin (2 θ))

\int cos^{3} (θ) d θ = sin (θ) - \frac{sin ^{3} ( θ )}{3}

= θ - 3 sin (θ) + \frac{3}{2} (θ + \frac{1}{2} sin (2 θ)) - (sin (θ) - \frac{sin ^{3} ( θ )}{3})

Vereinfache

θ - 3 sin (θ) + \frac{3}{2} (θ + \frac{1}{2} sin (2 θ)) - (sin (θ) - \frac{sin ^{3} ( θ )}{3}) : \frac{5}{2} θ + \frac{3}{4} sin (2 θ) - 4 sin (θ) + \frac{1}{3} sin^{3} (θ)

= \frac{5}{2} θ + \frac{3}{4} sin (2 θ) - 4 sin (θ) + \frac{1}{3} sin^{3} (θ)

Füge eine Konstante zur Lösung hinzu

= \frac{5}{2} θ + \frac{3}{4} sin (2 θ) - 4 sin (θ) + \frac{1}{3} sin^{3} (θ) + C

Graph

Beliebte Beispiele

integral of (2x^2+3x+7)

\int (2 x^{2} + 3 x + 7) d x

derivative f(x)=7x^3e^{-x}

d er i v a t i v e f (x) = 7 x^{3} e^{- x}

integral of e^{2x}sin(e^{2x})

\int e^{2 x} sin (e^{2 x}) d x

derivative y=10000+30xe^{x/(800)}

d er i v a t i v e y = 10000 + 30 x e^{\frac{x}{800}}

fläche x+2y=2,y-x=1,2x+y=7

a re a x + 2 y = 2, y - x = 1, 2 x + y = 7