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

The integral of (csc(θ))/(acsc(θ)-asin(θ)) is 1/a tan(θ)+C

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integral of (csc(θ))/(acsc(θ)-asin(θ))

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

\frac{1}{a} tan (θ) + C

Solution steps

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

\frac{csc ( θ )}{a csc ( θ ) - a sin ( θ )} = \frac{1}{a} \frac{csc ( θ )}{csc ( θ ) - sin ( θ )}

= \int \frac{1}{a} \cdot \frac{csc ( θ )}{csc ( θ ) - sin ( θ )} d θ

Take the constant out:

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

= \frac{1}{a} \cdot \int \frac{csc ( θ )}{csc ( θ ) - sin ( θ )} d θ

Simplify

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

= \frac{1}{a} \cdot \int sec^{2} (θ) d θ

Use the common integral:

\int sec^{2} (θ) d θ = tan (θ)

= \frac{1}{a} tan (θ)

Add a constant to the solution

= \frac{1}{a} tan (θ) + C

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What is the integral of (csc(θ))/(acsc(θ)-asin(θ)) ?
The integral of (csc(θ))/(acsc(θ)-asin(θ)) is 1/a tan(θ)+C