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Math Help - trig integral 1/(1-sin(x)cos(x))

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    trig integral 1/(1-sin(x)cos(x))

    \int_{0}^{\pi /2}{\frac{dx}{1-\sin (x)\cos (x)}}

    i saw this integral in a previous post and i'm curious on how to do it. i tried a wierstrauss substitution but that seems to make the problem extremely complicated. i also tried rewriting as {\frac{dx}{1-{\frac{1}{2}}\sin (2x)}} but i'm not sure on what to do next.

    what would be the best way to tackle this integral?
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  2. #2
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    Quote Originally Posted by oblixps View Post
    \int_{0}^{\pi /2}{\frac{dx}{1-\sin (x)\cos (x)}}

    i saw this integral in a previous post and i'm curious on how to do it. i tried a wierstrauss substitution but that seems to make the problem extremely complicated. i also tried rewriting as {\frac{dx}{1-{\frac{1}{2}}\sin (2x)}} but i'm not sure on what to do next.

    what would be the best way to tackle this integral?
    Make the substitution

    u = \tan{\frac{x}{2}} so du = \frac{1}{2}\sec^2{\frac{x}{2}}

    and make appropriate transformations for \sin{x} and \cos{x}.
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    yes that is the Weierstrass substitution. i tried that but it seemed to make the problem very complicated. is that the best way to do this problem?
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    Quote Originally Posted by oblixps View Post
    yes that is the Weierstrass substitution. i tried that but it seemed to make the problem very complicated. is that the best way to do this problem?


    consider  1 = \sin^2(x) + \cos^2(x)

     \int \frac{dx}{ 1- \sin(x)\cos(x) }

     =  \int \frac{dx}{\sin^2(x) - \sin(x)\cos(x) + \cos^2(x) }

     = \int \frac{ \sec^2(x)~dx }{ \tan^2(x)  - \tan(x) + 1}

    Sub.  t = \tan(x) , dt = \sec^2(x)dx

     I = \int_0^{\infty} \frac{dt}{t^2 - t + 1}
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    Multiply numerator and denominator by 1 + sinxcosx and then substitute cos^2x = 1 - sin^2x and you should be able to work your way from there.

    1 - sin^4x = (1 - sin^2x)(1 + sin^2x) and then use partial fractions (just an alternate way of solving).

    1 + sin^2x = (1 + isinx)(1 - isinx) and then use partial fractions again.

    [Next time, make all of your ideas into one single post. -K.]
    Last edited by Krizalid; January 26th 2010 at 08:19 AM.
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    [IMG]file:///C:/Users/Guest/Desktop/properties%20of%20definite%20integrals.bmp[/IMG]
    i was wondering whether some property of definite integral could be used.
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