integral of (1+cot(2θ))^2

解答

\int (1 + cot (2 θ))^{2} d θ

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

求解步骤

\int (1 + cot (2 θ))^{2} d θ

乘开

(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

使用积分加法定则:

\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)

化简

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

解答补常数

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

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