What is the integral of (1+cot(2θ))^2 ?

The integral of (1+cot(2θ))^2 is -1/2 cot(2θ)+ln|sin(2θ)|+C

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integral of (1+cot(2θ))^2

\int (1 + cot (2 θ))^{2} d θ

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

Solution steps

\int (1 + cot (2 θ))^{2} d θ

Expand

(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

Apply the Sum Rule:

\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)

Simplify

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

Add a constant to the solution

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

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What is the integral of (1+cot(2θ))^2 ?
The integral of (1+cot(2θ))^2 is -1/2 cot(2θ)+ln|sin(2θ)|+C