Question

What is the integral of `sec(x)`?

Answer 1

`intsecxdx=ln|secx+tanx|+C`

Explanation:

Integrating the secant requires a bit of manipulation.

Multiply `secx` by `(secx+tanx)/(secx+tanx)`, which is really the same as multiplying by `1.` Thus, we have

`int((secx(secx+tanx))/(secx+tanx))dx`

`int(sec^2x+secxtanx)/(secx+tanx)dx`

Now, make the following substitution:

`u=secx+tanx`

`du=(secxtanx+sec^2x)dx=(sec^2x+secxtanx)dx`

We see that `du` appears in the numerator of the integral, so we may apply the substitution:

`int(du)/u=ln|u|+C`

Rewrite in terms of `x` to get

`intsecxdx=ln|secx+tanx|+C`

This is an integral worth memorizing.