How do you integrate arcsec(x)?

1 Answer
Mar 9, 2015

Method: To integrate arcsec(x), substitution, then integrate by parts.

You'll also need secudu, which can be done by substitution and partial fractions.
Here's a nice explanation: http://socratic.org/questions/what-is-the-integral-of-sec-x .

Details:arcsec(x)dx

Let y=arcsec(x), so x=secy and dx=secytanydy.

With this substitution, the integral becomes:

ysecytanydy.

Integrate this by parts:
Let u=y and dv=secytanydy.
Then du=dy and v=secy.

uvvdu=ysecysecydy
=ysecyln|secy+tany|+C.

With y=arcsec(x) we get x=secy, and tany=x21.

The integral becomes:

arcsecxdx=(arcsec(x))xlnx+x21+C.

This is more easily read is we write it as:

x(arcsec(x))lnx+x21+C.