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Math Help - Integration

  1. #1
    MHF Contributor Amer's Avatar
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    Integration

    \displaystyle { \int \frac{1}{2-\sin x} dx  }

    any hints how I can integrate it
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  2. #2
    Super Member
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    Sub.  x = \frac{\pi}{2} - 2t

    we have


     \int \frac{dx}{2 - \sin(x) } = - 2 \int \frac{dt}{ 2 - \cos(2t) }


     = -2 \int \frac{dt}{ 1 + (1 - \cos(2t) )}

     = -2 \int \frac{dt}{1 + 2\sin^2(t) }

     = -2 \int \frac{dt}{ \cos^2(t) + 3\sin^2(t) }

     = -2 \int \frac{ \sec^2(t) ~dt }{ 1 + 3 \tan^2(t) }

     = - \frac{2}{\sqrt{3} } \tan^{-1}[ \sqrt{3} \tan(t) ] + C

    Then back substitution .
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  3. #3
    MHF Contributor Amer's Avatar
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    Thank you very much I appreciate that
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