How do you find the integral of 1cosx?

1 Answer
Apr 30, 2016

ln|secx+tanx|+C

Explanation:

Note that 1cosx=secx. Thus, we see that

1cosxdx=secxdx

This is an important trigonometric identity:

secxdx=ln|secx+tanx|+C


If you want to know how to find secxdx, the process is unintuitive, but I've outlined it below:

secxdx=secx(secx+tanx)secx+tanxdx=secxtanx+sec2xsecx+tanxdx

Now, use substitution:

u=secx+tanx du=(secxtanx+sec2x)dx

Note that these are the fraction's numerator and denominator.

secxtanx+sec2xsecx+tanxdx=duu=ln|u|+C

Since u=secx+tanx, we obtain

secxdx=ln|secx+tanx|+C