# The integral of sec (x)

The integral of sec(x) is given by:

∫ sec(x) dx = ln |sec(x) + tan(x)| + C

where C is a constant. This result can be derived using the following steps:

- Recall that sec(x) = 1/cos(x), so sec'(x) = tan(x)/cos(x).
- Use the substitution u = cos(x) to rewrite the integral as follows:

∫ sec(x) dx = ∫ du/u

- Integrate both sides:

∫ sec(x) dx = ∫ du/u = ln |u| + C = ln |cos(x)| + C

- Use the identity cos(x) = sec(x) – tan(x) to rewrite the integral as follows:

∫ sec(x) dx = ln |cos(x)| + C = ln |sec(x) – tan(x)| + C

- Use the identity sec(x) + tan(x) = sec(x) / cos(x) = 1/cos(x) = sec(x) to rewrite the integral as follows:

∫ sec(x) dx = ln |sec(x) – tan(x)| + C = ln |sec(x) + tan(x)| + C

This is the final form of the integral of sec(x). It can be used to calculate the indefinite integral of any function that can be expressed in terms of sec(x).

Here are some examples of how to use the integral of sec(x) to calculate indefinite integrals:

Example 1:

Find the indefinite integral of sec(x):

∫ sec(x) dx = ln |sec(x) + tan(x)| + C

Example 2:

Find the indefinite integral of 3*sec(x):

∫ 3*sec(x) dx = ∫ sec(x) dx * 3 = 3 * ln |sec(x) + tan(x)| + C

Example 3:

Find the indefinite integral of sec^2(x):

∫ sec^2(x) dx = tan(x) + C