What is the integral of (csc(θ))/(csc(θ)-sin(θ)) ?

The integral of (csc(θ))/(csc(θ)-sin(θ)) is tan(θ)+C

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integral of (csc(θ))/(csc(θ)-sin(θ))

\int \frac{csc ( θ )}{csc ( θ ) - sin ( θ )} d θ

tan (θ) + C

Solution steps

\int \frac{csc ( θ )}{csc ( θ ) - sin ( θ )} d θ

Simplify

\frac{csc ( θ )}{csc ( θ ) - sin ( θ )} : sec^{2} (θ)

= \int sec^{2} (θ) d θ

Use the common integral:

\int sec^{2} (θ) d θ = tan (θ)

= tan (θ)

Add a constant to the solution

= tan (θ) + C

What is the integral of (csc(θ))/(csc(θ)-sin(θ)) ?
The integral of (csc(θ))/(csc(θ)-sin(θ)) is tan(θ)+C