Expert: Paul Klarreich - 1/21/2008

Question
QUESTION: Using trig Substitution:
I am to integrate 1/sqr((e^2x)-1)
Because of this form
I use the substitution x=a sec(t)   
So e^x = sec(t)
And e^x dx = sec(t) tan(t) dt

This leads me to the integral sec(t) tan(t) / sqr((sec^2(t)-1))
After canceling the tan(t)  I have the integral sec(t) dt

I come up with the answer

ln (sec(t) + tan(t)) + C

after solving the triangle:

ln (e^x + (sqrt((e^2x)-1) / 1) + C

By book has Arcsec e^x  + C


Whose Right ?

By the way, this problem is from a chapter on Trig Substitutions.
Integrating Inverse trig Functions versus Trig Sustitution
Can be very confusing.

Thanks
Mark


ANSWER: Questioner:   Mark
Category:  Calculus
Private:  No
 
Subject:  Trig Substitution
Question:  Using trig Substitution:
I am to integrate 1/sqr((e^2x)-1)
Because of this form
I use the substitution x=a sec(t)   
So e^x = sec(t)
And e^x dx = sec(t) tan(t) dt

This leads me to the integral sec(t) tan(t) / sqr((sec^2(t)-1))

************** Error is in that line.  The sec(t) should have cancelled.


After canceling the tan(t)  I have the integral sec(t) dt

I come up with the answer

ln (sec(t) + tan(t)) + C

after solving the triangle:

ln (e^x + (sqrt((e^2x)-1) / 1) + C

By book has Arcsec e^x  + C


Whose Right ?

By the way, this problem is from a chapter on Trig Substitutions.
Integrating Inverse trig Functions versus Trig Sustitution
Can be very confusing.

Thanks
Mark
.......................................

Hi, Mark,

Sorry, but the book is right.  You got started OK, but lost something on the way. It goes like this:


{     dx
| --------------
} sqrt(e^2x - 1)

Let  e^x = sec t  << you got this
e^x dx = sec t tan t dt  << this, too.
sec t dx = sec t tan t dt  << because e^x = sec t

dx = tan t dt  < after cancelling sec t, which YOU FORGOT.

{     tan t dt
| --------------
} sqrt(sec^2 t - 1)

{     tan t dt
| --------------
} sqrt(tan^2 t)  << from a trig identity, sec^2 - 1 = tan^2

{    tan t dt
| --------------
}    tan t

{
| t dt   ( Wow!!)
}

= t = arcsec e^x, because sec t = e^x.


---------- FOLLOW-UP ----------

QUESTION: I had let e^x dx = sec(t) tan(t) dt
But there was no extra e^x in the problem.
So we had to get rid of it by letting e^x = sec(t) in the line e^x dx = sec(t) tan(x) dt and then cancelling the sec(t)
Do I understand correctly?

Answer
Questioner:   Mark
Category:  Calculus
Private:  No
 
---------- FOLLOW-UP ----------

QUESTION: I had let e^x dx = sec(t) tan(t) dt

>> No, you didn't.  You let e^x = sec t, which was correct.
Then you GOT

e^x dx = sec t tan t dt.

But there was no extra e^x in the problem.
So we had to get rid of it by letting (NOT LETTING, substituting.  You had ALREADY let e^x = sec t.)

e^x = sec(t) in the line e^x dx = sec(t) tan(x) dt and then cancelling the sec(t)
Do I understand correctly?

THAT'S IT, exactly.