Integral involving hyperbolic functions

  • Thread starter DryRun
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  • #1
DryRun
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Homework Statement
Find [tex]\int \frac{x}{\sqrt{2x^2-2x+1}}\,dx[/tex]

The attempt at a solution

First, i complete the square for the quadratic expression:
[tex]2x^2-2x+1=2((x-\frac{1}{2})^2+\frac{1}{4})[/tex]
[tex]\int \frac{x}{\sqrt{2x^2-2x+1}}\,dx=\int \frac{x}{\sqrt 2 \sqrt{(x-\frac{1}{2})^2+\frac{1}{4}}}\,dx[/tex]

Using substitution: Let [itex](x-\frac{1}{2})=\frac{1}{2}\sinh u[/itex]

I eventually get this expression, after integrating:
[tex]\frac{1}{2\sqrt 2}(\cosh (\sinh^{-1}(2x-1)+\sinh^{-1}(2x-1))[/tex]

But i don't know how to simplify this any further, without getting a complicated answer. The final answer is: [itex]\frac{\sinh^{-1}1}{\sqrt 2}[/itex]
 
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Answers and Replies

  • #2
tiny-tim
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hi sharks! :smile:

(you haven't said what the limits are)
I eventually get this expression, after integrating:
[tex]\frac{1}{2\sqrt 2}(\cosh (\sinh^{-1}(2x-1)+\sinh^{-1}(2x-1))[/tex]

But i don't know how to simplify this any further

cosh(sinh-1x) = √(x2 + 1) :wink:

(btw, you could have simplified the original integrand by rewriting the numerator as x - 1/2 + 1/2 :wink:)
 
  • #3
DryRun
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Hi tiny-tim! :smile:

Congratulations on your PF award!

Doh! I forgot to include the limits.

[tex]\int^1_0 \frac{x}{\sqrt{2x^2-2x+1}}\,dx[/tex]

If i evaluate over the limits:
[tex]\frac{1}{2\sqrt 2}(\cosh (\sinh^{-1}(2x-1)+\sinh^{-1}(2x-1))[/tex]
Indeed, i get the answer!

I am trying to understand your suggestion:
[tex]\cosh (\sinh^{-1}x) = \sqrt{(x^2 + 1)}[/tex]
It's too complicated to prove? I can't get the R.H.S. from the L.H.S.

Here is where i've reached...
[tex]\frac{x^2+x\sqrt{x^2+1}+1}{x+\sqrt{x^2+1}}[/tex]
 
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  • #4
tiny-tim
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I am trying to understand your suggestion:
[tex]\cosh (\sinh^{-1}x) = \sqrt{(x^2 + 1)}[/tex]

if y = sinh-1x

then x = sinhy,

so √(x2 + 1) = √(sinh2y + 1) = coshy = cosh(sinh-1x) :smile:

(if you think about it, this is obvious! :wink:)
 
  • #5
DryRun
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I have also tried to solve this problem using your other suggestion - rewriting the numerator as x - 1/2 + 1/2, and i was able to get the answer.

Thank you for your insightful suggestions, tiny-tim. :smile:
 

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