What is the integral of (sec(θ))/(asec(θ)-acos(θ)) ?

The integral of (sec(θ))/(asec(θ)-acos(θ)) is -1/a cot(θ)+C

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integral of (sec(θ))/(asec(θ)-acos(θ))

\int \frac{sec ( θ )}{a sec ( θ ) - a cos ( θ )} d θ

- \frac{1}{a} cot (θ) + C

Solution steps

\int \frac{sec ( θ )}{a sec ( θ ) - a cos ( θ )} d θ

\frac{sec ( θ )}{a sec ( θ ) - a cos ( θ )} = \frac{1}{a} \frac{sec ( θ )}{sec ( θ ) - cos ( θ )}

= \int \frac{1}{a} \cdot \frac{sec ( θ )}{sec ( θ ) - cos ( 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{sec ( θ )}{sec ( θ ) - cos ( θ )} d θ

Rewrite using trig identities

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

Rewrite using trig identities

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

Use the common integral:

\int csc^{2} (θ) d θ = - cot (θ)

= \frac{1}{a} (- cot (θ))

Simplify

= - \frac{1}{a} cot (θ)

Add a constant to the solution

= - \frac{1}{a} cot (θ) + C

What is the integral of (sec(θ))/(asec(θ)-acos(θ)) ?
The integral of (sec(θ))/(asec(θ)-acos(θ)) is -1/a cot(θ)+C