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Alternate, superior solution to an integral

  1. Nov 21, 2005 #1

    GCT

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    Hi all, consider the integral
    [tex]I= \int \frac{dx}{cos^2 x( \sqrt{ cos^2 x - cos^2 a} ) } [/tex]
    I've solved it with three substitutions (don't have time to post it at the moment) but was wondering if anyone here can provide a superior, more simplistic solution; e.g. one substitution with proper algebraic simplification.
     
  2. jcsd
  3. Nov 21, 2005 #2

    GCT

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    the first substitution was [tex]u=cos \theta [/tex] the second was [tex]u=sec w [/tex], and final substitution was [tex]z = sin w [/tex], had to use some trig identities in between and focus in on the simplifications. My current solution is (I'll still have yet to review it close up for dumb mistakes)

    [tex] I= \frac{i}{cos a} [ \frac{( \frac{2(cos^2 x -1)}{cosx}) ^2 ln( \frac{2(cos^2 x -1)}{cosx} ) + ( \frac{2(cos^2 x -1)}{cosx} ) ^4 -1 }{2 ( \frac{2(cos^2 x -1)}{cosx} ) ^2 } ~] + C [/tex]

    I'm having some trouble with latex

    I = (i/cosa)[2{(cos^2 x -1)/(cosx)}^2 ln {(cos^2 x -1)/(cosx)} + {(cos^2 x -1)/(cosx))^2}^4 - 1]/[2 {(cos^2 x -1)/(cosx)}^2] + C
     
    Last edited: Nov 21, 2005
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