Find the integral of sec x.

To find the integral of sec(x), we can use a technique called substitution. Let’s set u = sec(x) + tan(x). Then, we can find the derivative of u with respect to x:

du/dx = sec(x) tan(x) + sec^2(x)

Notice that the second term in this expression is just u^2, since u = sec(x) + tan(x). So we can rewrite this as:

du/dx = u^2 + sec(x) tan(x)

Now we can rearrange this equation to solve for the integral of sec(x):

du / (u^2 + sec(x) tan(x)) = dx

We can recognize the denominator on the left-hand side as the derivative of the arctangent function. So we can substitute u = tan(z) and du = sec^2(z) dz, giving:

dz / (1 + tan^2(z)) = dx

This is the derivative of the arctangent function, so we can integrate both sides:

arctan(u) = integral of dz / (1 + tan^2(z)) = arctan(tan(z)) + C

Substituting back for z and u gives:

arctan(sec(x) + tan(x)) = integral of sec(x) dx + C

Therefore, the integral of sec(x) is:

integral of sec(x) dx = arctan(sec(x) + tan(x)) + C

where C is the constant of integration.