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Chapter 4: Problem 23

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

Short Answer

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

Step by step solution

01

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 \( \int \frac{du}{u} \)
02

Solve the Integral

The integral now reduces to the form that we know easily: \( \int \frac{du}{u} = ln |u| + C \)
03

Substitute back original variable

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

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