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

The integral of xcsc^2(2x) is -1/2 xcot(2x)+1/4 ln|sin(2x)|+C

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integral of xcsc^2(2x)

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

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

Solution steps

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

Apply Integration By Parts

= - \frac{1}{2} x cot (2 x) - \int - \frac{1}{2} cot (2 x) d x

\int - \frac{1}{2} cot (2 x) d x = - \frac{1}{4} ln ∣ sin (2 x) ∣

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

Simplify

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

Add a constant to the solution

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

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