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Math Help - How to integrate dx/(5+4sin x)

  1. #1
    Junior Member silvercats's Avatar
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    How to integrate dx/(5+4sin x)

    dx/(5+4sin x)
    step by step explanation would be appreciated
    thanks
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  2. #2
    Super Member girdav's Avatar
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    When you don't have any idea you can put t=\tan \dfrac x2.
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  3. #3
    Junior Member silvercats's Avatar
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    yeah i took t=tan (x/2) and  dt=secē( x/2)

    But i don't know what to do after that :/
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  4. #4
    MHF Contributor chisigma's Avatar
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    The complete substitution is...

    \displaystyle t= \tan \frac{x}{2}

    \displaystyle dx= \frac{2\ dt}{1+t^{2}}

    \displaystyle \sin x= \frac{2\ t}{1+t^{2}}

    \displaystyle \cos x= \frac{1-t^{2}}{1+t^{2}}

    Kind regards

    \chi \sigma
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  5. #5
    Junior Member silvercats's Avatar
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    how did that sin x = 2t/(1+tē) thing come here?i don't know about that trigonometry identity.somebody want to prove it to me?
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  6. #6
    MHF Contributor chisigma's Avatar
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    \displaystyle \sin x = 2 \sin \frac{x}{2}\ \cos \frac{x}{2} = \frac{2 \sin \frac{x}{2}\ \cos \frac{x}{2}}{\sin^{2} \frac{x}{2} + \cos^{2} \frac{x}{2}} = \frac{2\ \tan \frac{x}{2}}{1+\tan^{2} \frac{x}{2}} = \frac{2 t}{1+t^{2}}

    Kind regards

    \chi \sigma
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  7. #7
    Junior Member silvercats's Avatar
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    i know that [maths]sin (2x) =2 sin x.cos x [/maths].
    but how is it  sin x =2 sin (x/2 ).cos (x/2) ?

    and [(2 sin (x/2) cos (x/2)]/[(sinē (x/2) + cosē (x/2)]= 2 Tan (x/2)/(1 + Tanē (x/2) ?
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  8. #8
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by silvercats View Post
    i know that [maths]sin (2x) =2 sin x.cos x [/maths].
    but how is it  sin x =2 sin (x/2 ).cos (x/2) ?

    and [(2 sin (x/2) cos (x/2)]/[(sinē (x/2) + cosē (x/2)]= 2 Tan (x/2)/(1 + Tanē (x/2) ?
    sin(2x) = 2~sin(x)~cos(x)

    Now let y = 2x, Thus x = y/2...



    For the other question, divide top and bottom by cos^2 \left ( \frac{x}{2} \right )

    -Dan
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  9. #9
    Junior Member silvercats's Avatar
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    thanks chisigma and topsquark !!!!!!
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