How do you determine if the improper integral converges or diverges int sec^2 x dx from negative 0 to pi?

1 Answer
Nov 4, 2016

The integral diverges.

Explanation:

Let I=int_0^p sec^2x dx

If the integral does converge then by symmetry we have :

I=2 int_0^(pi/2) sec^2x dx

And then as the integrand is invalid at pi/2 the formal definition would be
I=2lim_(nrarr pi/2) int_0^n sec^2x dx

:. I=2lim_(nrarr pi/2) [tan x]_0^n

:. I=2lim_(n rarr pi/2) (tann - tan0)
:. I=2lim_(n rarr pi/2) tann
:. I rarr oo , as tann rarr oo as n rarr pi/2

Hence, The integral diverges.