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Integral help

  • #1

Homework Statement


trying to integrate this:
[tex]\int^{\theta}_{\theta_{0}} \sqrt{\frac{1-cos(\theta)}{cos(\theta_{0}) - cos(\theta)}d\theta[/tex]


Homework Equations



My book tells me to let theta = pi - 2gamma and then simplify from there but I'm just not seeing that ! any hints? is there a trig identity that I'm missing ? (the only thing I see is to change the sign of the cosines but I don't see where that gets me).

The Attempt at a Solution

 

Answers and Replies

  • #2
dextercioby
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Your integrand kind of blows up in the lower limit. Make the following substitutions and see what you get

[tex] \cos \theta_0 = A \, , \, \tan\frac{\theta}{2} = x [/tex]

Do you get a simplification ?
 
  • #3
SammyS
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Get the dθ out of the radicand.

[tex]\int^{\theta}_{\theta_{0}} \sqrt{\frac{1-\cos(\theta)}{\cos(\theta_{0}) - \cos(\theta)}}\ d\theta[/tex]

What do you get if you use the hint, [tex]\text{Let }\theta=\pi-2\,\gamma\,?[/tex]

[tex]\cos(\theta)=\cos(\pi-2\,\gamma)=\cos(\pi)\cos(2\,\gamma)+\sin(\pi)\sin(2\,\gamma)=\quad?[/tex]

Then you have a choice for [tex]\cos(2\,\gamma).\quad\quad\cos(2x)=2\cos^2(x)-1=1-2\sin^2(x)[/tex]
 
  • #4
Btw... the upper limit should be pi...

And I forgot about my double angle formula... just now working through simplifying with that

[tex]
\cos(\theta)=\cos(\pi-2\,\gamma)=\cos(\pi)\cos(2\,\gamma)+\sin(\pi)\sin( 2\,\gamma)= -cos(2\gamma)
[/tex]
 

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