What is the integral of xcsc^2(3x) ?

The integral of xcsc^2(3x) is -1/3 xcot(3x)+1/9 ln|sin(3x)|+C

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integral of xcsc^2(3x)

\int x csc^{2} (3 x) d x

- \frac{1}{3} x cot (3 x) + \frac{1}{9} ln ∣ sin (3 x) ∣ + C

Solution steps

\int x csc^{2} (3 x) d x

Apply Integration By Parts

: - \frac{1}{3} x cot (3 x) - \int - \frac{1}{3} cot (3 x) d x

= - \frac{1}{3} x cot (3 x) - \int - \frac{1}{3} cot (3 x) d x

\int - \frac{1}{3} cot (3 x) d x = - \frac{1}{9} ln ∣ sin (3 x) ∣

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

Simplify

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

Add a constant to the solution

= - \frac{1}{3} x cot (3 x) + \frac{1}{9} ln ∣ sin (3 x) ∣ + C

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What is the integral of xcsc^2(3x) ?
The integral of xcsc^2(3x) is -1/3 xcot(3x)+1/9 ln|sin(3x)|+C