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Beliebt Rechnen >

integral of (cos(θ))/(5-3cos(θ))

Lösung

\int \frac{cos ( θ )}{5 - 3 cos ( θ )} d θ

Lösung

- \frac{1}{3} θ + \frac{5}{6} arctan (2 tan (\frac{θ}{2})) + C

Schritte zur Lösung

\int \frac{cos ( θ )}{5 - 3 cos ( θ )} d θ

Lange Division

\frac{cos ( θ )}{5 - 3 cos ( θ )} : - \frac{1}{3} + \frac{5}{3 ( - 3 cos ( θ ) + 5 )}

= \int - \frac{1}{3} + \frac{5}{3 ( - 3 cos ( θ ) + 5 )} 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 \frac{1}{3} d θ + \int \frac{5}{3 ( - 3 cos ( θ ) + 5 )} d θ

\int \frac{1}{3} d θ = \frac{1}{3} θ

\int \frac{5}{3 ( - 3 cos ( θ ) + 5 )} d θ = \frac{5}{6} arctan (2 tan (\frac{θ}{2}))

= - \frac{1}{3} θ + \frac{5}{6} arctan (2 tan (\frac{θ}{2}))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} θ + \frac{5}{6} arctan (2 tan (\frac{θ}{2})) + C

Graph

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