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Math Help - integral

  1. #1
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    integral

    Hi all

    The problem is: Show that the value of the integral

    int_{0,2} (375x^5)(x^2 +1)^(-4) dx is 2^n for some integer n.

    Thank you
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  2. #2
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    Quote Originally Posted by storchfire1X View Post
    Hi all

    The problem is: Show that the value of the integral

    int_{0,2} (375x^5)(x^2 +1)^(-4) dx is 2^n for some integer n.

    Thank you
    Let us ignore the 375 factor for now.
    \int_0^2 \frac{x^5}{(x^2+1)^4} dx = \int_0^2 \frac{x^5}{(x^2+1)^3} \cdot \frac{1}{x^2+1} dx

    Let t=\tan^{-1} x thus, t'=\frac{1}{x^2+1}.
    And \sin t = x and \cos t = \frac{1}{\sqrt{x^2+1}}, hence, x^5 = \sin^5 t \mbox{ and }\frac{1}{(x^2+1)^3} = \cos^6 t

    Substitution theorem yields,
    \int_0^{\tan^{-1} 2} \frac{\sin^5 t}{\cos^6 t} dt = \int_0^{\tan^{-1} 2} \frac{(1-\cos^2 t)^2 \sin t}{\cos^6 t}dt

    Let s=\cos t then s'=-\sin t

    The substitution rule says that,
    -\int_1^{2/\sqrt{5}} \frac{(1-s^2)^2}{s^6} ds = \int_{2/\sqrt{5}}^1 \frac{1-2s^2+s^4}{s^6}ds
    Now show that multiplication by 375 will make this an exponent of 2. The rest is more computational.
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