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Beliebt Rechnen >

integral of (1+cot(2θ))^2

Lösung

\int (1 + cot (2 θ))^{2} d θ

Lösung

- \frac{1}{2} cot (2 θ) + ln ∣ sin (2 θ) ∣ + C

Schritte zur Lösung

\int (1 + cot (2 θ))^{2} d θ

Multipliziere aus

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

= \int 1 + 2 cot (2 θ) + cot^{2} (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

= \int 1 d θ + \int 2 cot (2 θ) d θ + \int cot^{2} (2 θ) d θ

\int 1 d θ = θ

\int 2 cot (2 θ) d θ = ln ∣ sin (2 θ) ∣

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

= θ + ln ∣ sin (2 θ) ∣ - θ - \frac{1}{2} cot (2 θ)

Vereinfache

= - \frac{1}{2} cot (2 θ) + ln ∣ sin (2 θ) ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} cot (2 θ) + ln ∣ sin (2 θ) ∣ + C

Graph

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