What is the integral of (csc^2(2x))/(4-cot(2x)) ?

The integral of (csc^2(2x))/(4-cot(2x)) is 1/2 ln|4-cot(2x)|+C

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integral of (csc^2(2x))/(4-cot(2x))

\int \frac{csc ^{2} ( 2 x )}{4 - cot ( 2 x )} d x

\frac{1}{2} ln ∣ 4 - cot (2 x) ∣ + C

Solution steps

\int \frac{csc ^{2} ( 2 x )}{4 - cot ( 2 x )} d x

Apply u-substitution

= \int \frac{1}{2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = 4 - cot (2 x)

= \frac{1}{2} ln ∣ 4 - cot (2 x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ 4 - cot (2 x) ∣ + C

What is the integral of (csc^2(2x))/(4-cot(2x)) ?
The integral of (csc^2(2x))/(4-cot(2x)) is 1/2 ln|4-cot(2x)|+C