What is the integral of sec x?

The integral of sec x is given by the formula:

∫ sec x dx = ln |sec x + tan x| + C

where C is the constant of integration. To derive this result, we can use a clever trick that involves multiplying and dividing by (sec x + tan x):

1. Start with the integral: ∫ sec x dx
2. Multiply and divide by (sec x + tan x):

∫ sec x dx = ∫ sec x (sec x + tan x)/(sec x + tan x) dx

3. This simplifies to:

∫ (sec² x + sec x tan x)/(sec x + tan x) dx

4. Now, let u = sec x + tan x. Then, the derivative du = (sec x tan x + sec² x) dx
5. Thus, we can rewrite the integral in terms of u: ∫ 1/u du
6. This evaluates to:

ln |u| + C = ln |sec x + tan x| + C

And that is how we arrive at the final answer for the integral of sec x. It’s a neat example of how using substitution can simplify seemingly complicated integrals!