Integrating the secant requires a bit of manipulation.
Multiply secxsecx by (secx+tanx)/(secx+tanx)secx+tanxsecx+tanx, which is really the same as multiplying by 1.1. Thus, we have
int((secx(secx+tanx))/(secx+tanx))dx∫(secx(secx+tanx)secx+tanx)dx
int(sec^2x+secxtanx)/(secx+tanx)dx∫sec2x+secxtanxsecx+tanxdx
Now, make the following substitution:
u=secx+tanxu=secx+tanx
du=(secxtanx+sec^2x)dx=(sec^2x+secxtanx)dxdu=(secxtanx+sec2x)dx=(sec2x+secxtanx)dx
We see that dudu appears in the numerator of the integral, so we may apply the substitution:
int(du)/u=ln|u|+C∫duu=ln|u|+C
Rewrite in terms of xx to get
intsecxdx=ln|secx+tanx|+C∫secxdx=ln|secx+tanx|+C
This is an integral worth memorizing.
