What is the integral of (9csc^2(3x))/(3-cot(3x)) ?

The integral of (9csc^2(3x))/(3-cot(3x)) is 3ln|3-cot(3x)|+C

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integral of (9csc^2(3x))/(3-cot(3x))

\int \frac{9 csc ^{2} ( 3 x )}{3 - cot ( 3 x )} d x

3 ln ∣ 3 - cot (3 x) ∣ + C

Solution steps

\int \frac{9 csc ^{2} ( 3 x )}{3 - cot ( 3 x )} d x

Take the constant out:

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

= 9 \cdot \int \frac{csc ^{2} ( 3 x )}{3 - cot ( 3 x )} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

= 9 \cdot \int \frac{1}{3 u} d u

Take the constant out:

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

= 9 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 9 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = 3 - cot (3 x)

= 9 \cdot \frac{1}{3} ln ∣ 3 - cot (3 x) ∣

Simplify

9 \cdot \frac{1}{3} ln ∣ 3 - cot (3 x) ∣ : 3 ln ∣ 3 - cot (3 x) ∣

= 3 ln ∣ 3 - cot (3 x) ∣

Add a constant to the solution

= 3 ln ∣ 3 - cot (3 x) ∣ + C

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What is the integral of (9csc^2(3x))/(3-cot(3x)) ?
The integral of (9csc^2(3x))/(3-cot(3x)) is 3ln|3-cot(3x)|+C