Expert: Paul Klarreich - 1/24/2008

Question
I am studying integration by trig substitution.
I am asked to integrate [sqrt((x^2) + a^2)] / x

I keep coming up with the same answer of

Sqrt((x^2) + a^2)            sqrt((x^2) + a^2)      a
------------------ + a ln[ -------------------- - -----]
      1                              x             x


Schaums Outline 4th edition Chapter 32 prob 64 provides


Sqrt((x^2)+a^2         a       sqrt((a^2) +x^2)   - a
------------------- +  --  ln[ ------------------------]
    1                 2       sqrt((a^2) + x^2)  + a



who’s Right?
Thanks
Mark


Answer
Questioner:   Mark
Category:  Calculus
Private:  No
 
Subject:  Trig Substitution
Question:  I am studying integration by trig substitution.
I am asked to integrate [sqrt((x^2) + a^2)] / x

I keep coming up with the same answer of

Sqrt((x^2) + a^2)            sqrt((x^2) + a^2)      a
------------------ + a ln[ -------------------- - -----]
     1                              x             x


Schaums Outline 4th edition Chapter 32 prob 64 provides


Sqrt((x^2)+a^2         a       sqrt((a^2) +x^2)   - a
------------------- +  --  ln[ ------------------------]
   1                  2       sqrt((a^2) + x^2)  + a



who’s Right?
Thanks

Mark
...........................
Hi, Mark,

This took a lot of working out. [I had to actually use real paper and ink for it.]

But I am afraid the book is right, again.  [Sorry, but you're 0 for 2 now.]  At least you wrote "who's" instead of "whose", so your grammar is improving.

Your integral:
{
| [sqrt((x^2 + a^2)]dx / x
}

Let x = a tan t,  dx = a sec^2 t dt
x^2 + a^2 = a^2 sec^2 t

{
| a sec t a sec^2 t dt / a tan t
}

{ a sec^3 t dt
| -----------
}   tan t


{ a cos t dt
| -----------
}  cos^3 sin t

{   a dt
| -----------
}  cos^2 t sin t

{ a sin t dt
| -----------
}  cos^2 t sin^2 t

Now let  c = cos t,  dc = - sin t dt

{    - dc
|a -----------
}  c^2 (1 - c^)

I had to use Partial fractions:

A        D       E      F          1
---- + ------- + ----- + ----- = -----------
c       c^2     1 + c   1 - c    ..........

Ac(1 - c^2) + D(1 - c^2) + Ec^2(1 - c) + Fc^2(1 + c)

Ac - Ac^3 + D - Dc^2 + Ec^2 - Ec^3 + Fc^2 + Fc^3


0: D = 1

1: A = 0,  

2: -D + E + F = 0

3: - A  - E + F = 0

3:     - E + F = 0  -->  E = F

   - 1 + E + F = 0
--------------------------
   - 1 + 2F = 0

F = 1/2 = E
----------------This is the decomp:-------------

{     -1             1        1   
|a[ ------- - 1/2[ ----- + -------- ]dc =
}      c^2         1 + c    1 - c  

                    
[1/c]  - 1/2 [ ln(1+c) - ln(1-c)] =   << there's an 'a' there, too.

sec t - 1/2 ln[(1 + c)/1 - c)]

sec t + 1/2 ln[(1 - c)/1 + c)]
                  
OK, now, that resolves to the book's answer.