- #1
ilikesoldat
- 9
- 1
Homework Statement
Show that the two formulas are equivalent
integral [sec x dx] = ln|sec x + tan x| + C
integral [sec x dx] = -ln|sec x - tan x| + C
Homework Equations
Pythagorean ID's?
Log rule of addition
The Attempt at a Solution
Well, I realized the formulas can only be equivalent if
-ln|sec x - tan x| + C = ln|sec x + tan x| + C
I put the constants on one side, ln on the other:
ln |sec x + tan x| + ln| sec x - tan x| = C
Then,
ln |(sec x + tan x) * (sec x - tan x)| = C
ln |(sec x)^2 - (tan x)^2| = C
Using pythagorean ID's,
ln |1| = C
But what now, 0 = C doesn't show the two identities are equivalent... am I also supposed to assume in
-ln|sec x - tan x| + C = ln|sec x + tan x| + C
that the constants are equivalent so that
ln |sec x + tan x| + ln| sec x - tan x| = 0 ?
Because then ln |1| = 0 ??
I thought that constants must be different and that C-C doesn't necessarily = 0 so you can't assume that..
Did I approach this whole thing wrong? Even if ln |1| = 0, how would that prove that
-ln|sec x - tan x| + C = ln|sec x + tan x| + C it simplifies to ln |1| = 0, but just because the simplified result is a true statement (possibly not because what if ln |1| = C does that prove the original equations are equivalent? or ...?