What is the integral of sec(x)?

1 Answer
Apr 18, 2018

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.