How do you integrate arcsec(x)?

1 Answer
Mar 9, 2015

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

You'll also need int secu du, 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:int arcsec(x) dx

Let y=arc sec(x), so x=secy and dx = secy tany dy.

With this substitution, the integral becomes:

inty secy tany dy.

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

u v - int v du=ysecy-intsecy dy
=ysecy-ln abs(secy+tany)+C.

With y=arc sec(x) we get x=secy, and tany=sqrt (x^2-1).

The integral becomes:

int arcsec x dx = (arc sec(x))x-ln abs(x+sqrt(x^2-1))+C.

This is more easily read is we write it as:

x (arc sec(x)) - ln abs(x+sqrt(x^2-1))+C.