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

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Answer 1 · 2016-11-04T23:47:17

The integral diverges.

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.