What is the integral of θ/(csc(3θ)) ?

The integral of θ/(csc(3θ)) is -1/3 θcos(3θ)+1/9 sin(3θ)+C

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integral of θ/(csc(3θ))

\int \frac{θ}{csc ( 3 θ )} d θ

- \frac{1}{3} θ cos (3 θ) + \frac{1}{9} sin (3 θ) + C

Solution steps

\int \frac{θ}{csc ( 3 θ )} d θ

Use the basic trigonometric identity:

\frac{1}{csc ( x )} = sin (x)

= \int θ sin (3 θ) d θ

Apply Integration By Parts

= - \frac{1}{3} θ cos (3 θ) - \int - \frac{1}{3} cos (3 θ) d θ

\int - \frac{1}{3} cos (3 θ) d θ = - \frac{1}{9} sin (3 θ)

= - \frac{1}{3} θ cos (3 θ) - (- \frac{1}{9} sin (3 θ))

Simplify

= - \frac{1}{3} θ cos (3 θ) + \frac{1}{9} sin (3 θ)

Add a constant to the solution

= - \frac{1}{3} θ cos (3 θ) + \frac{1}{9} sin (3 θ) + C

What is the integral of θ/(csc(3θ)) ?
The integral of θ/(csc(3θ)) is -1/3 θcos(3θ)+1/9 sin(3θ)+C