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Beliebt Rechnen >

integral of cos^4(θ)sin^2(θ)

Lösung

\int cos^{4} (θ) sin^{2} (θ) d θ

Lösung

\frac{3}{8} θ - \frac{3}{32} sin (4 θ) + \frac{19}{24} cos^{3} (θ) sin (θ) - \frac{1}{6} cos^{5} (θ) sin (θ) - \frac{5}{32} (2 θ + sin (2 θ)) + C

Schritte zur Lösung

\int cos^{4} (θ) sin^{2} (θ) d θ

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int cos^{4} (θ) (1 - cos^{2} (θ)) d θ

Multipliziere aus

cos^{4} (θ) (1 - cos^{2} (θ)) : cos^{4} (θ) - cos^{6} (θ)

= \int cos^{4} (θ) - cos^{6} (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 cos^{4} (θ) d θ - \int cos^{6} (θ) d θ

\int cos^{4} (θ) d θ = cos^{3} (θ) sin (θ) + \frac{3}{8} (θ - \frac{1}{4} sin (4 θ))

\int cos^{6} (θ) d θ = \frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ)))

= cos^{3} (θ) sin (θ) + \frac{3}{8} (θ - \frac{1}{4} sin (4 θ)) - (\frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ))))

Vereinfache

cos^{3} (θ) sin (θ) + \frac{3}{8} (θ - \frac{1}{4} sin (4 θ)) - (\frac{sin ( θ ) cos ^{5} ( θ )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (θ) sin (θ) + \frac{3}{8} (θ + \frac{1}{2} sin (2 θ)))) : \frac{3}{8} θ - \frac{3}{32} sin (4 θ) + \frac{19}{24} cos^{3} (θ) sin (θ) - \frac{1}{6} cos^{5} (θ) sin (θ) - \frac{5}{32} (2 θ + sin (2 θ))

= \frac{3}{8} θ - \frac{3}{32} sin (4 θ) + \frac{19}{24} cos^{3} (θ) sin (θ) - \frac{1}{6} cos^{5} (θ) sin (θ) - \frac{5}{32} (2 θ + sin (2 θ))

Füge eine Konstante zur Lösung hinzu

= \frac{3}{8} θ - \frac{3}{32} sin (4 θ) + \frac{19}{24} cos^{3} (θ) sin (θ) - \frac{1}{6} cos^{5} (θ) sin (θ) - \frac{5}{32} (2 θ + sin (2 θ)) + C

Graph

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