Question

Find the indefinite integral. $$ \int \frac{\sec x \tan x}{\sec x-1} d x $$

Short answer

The indefinite integral of the given function is \( ln |sec x - 1| + C \)

Step-by-step solution

Determine Suitable Substitution

Let's do a substitution: let \(u = sec x -1\). Then differential of \(u\) is given by \(du = sec x \cdot tan x dx \). This substitution simplifies the integrand to \( \int \frac{du}{u} \)

Solve the Integral

The integral now reduces to the form that we know easily: \( \int \frac{du}{u} = ln |u| + C \)

Substitute back original variable

Substitute back the original variable in place of \(u\) to get the final answer as \(ln |sec x - 1| + C \)