FP2 Hyperbolic Functions

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  1. Horchata's Avatar
    1 01-06-2012  22:56
    • Adored and Respected Member
    • Posts: 581
    FP2 Hyperbolic Functions
    Hi,

    I'm trying to do this past paper question and I don't understand how to get to the second line of the solution from the mark scheme:


    Click image for larger version.  Name: HFquestion.jpg  Views: 26  Size: 17.3 KB  ID: 153302

    I understand that there has been integration by parts but I specifically don't know how it is known that

    \frac{2x}{\sqrt{9 + 4x^2}} integrates to 0.5\sqrt{9 + 4x^2}

    Is there a special rule that I'm missing? I've seen a lot of similar solutions where the solution is a fraction of the root, but I don't understand how they do it in one step.

    Thanks in advance to anyone who replies
  2. raheem94's Avatar
    2 01-06-2012  23:01
    • TSR Demigod
    • Posts: 5,512
    Re: FP2 Hyperbolic Functions
    (Original post by Horchata)
    Hi,

    I'm trying to do this past paper question and I don't understand how to get to the second line of the solution from the mark scheme:


    Click image for larger version.  Name: HFquestion.jpg  Views: 26  Size: 17.3 KB  ID: 153302

    I understand that there has been integration by parts but I specifically don't know how it is known that

    \frac{2x}{\sqrt{9 + 4x^2}} integrates to 0.5\sqrt{9 + 4x^2}

    Is there a special rule that I'm missing? I've seen a lot of similar solutions where the solution is a fraction of the root, but I don't understand how they do it in one step.

    Thanks in advance to anyone who replies
    \displaystyle \int \frac{2x}{\sqrt{9 + 4x^2}} \ dx = \int 2x(9+4x^2)^{- \frac12} \ dx

    This is of the form, \displaystyle \int kf'(x) [f(x)]^n \ dx = [f(x)]^{n+1}

    So if you differentiate u = (9+4x^2)^{\frac12} , you can find the answer.
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  3. Horchata's Avatar
    3 01-06-2012  23:08
    • Adored and Respected Member
    • Posts: 581
    Re: FP2 Hyperbolic Functions
    (Original post by raheem94)
    \displaystyle \int \frac{2x}{\sqrt{9 + 4x^2}} \ dx = \int 2x(9+4x^2)^{- \frac12} \ dx

    This is of the form, \displaystyle \int kf'(x) [f(x)]^n \ dx = [f(x)]^{n+1}

    So if you differentiate u = (9+4x^2)^{\frac12} , you can find the answer.
    OOHHH, should have seen that. Thanks so much!
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