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Math Help - Integration using Substitution

  1. #1
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    Integration using Substitution

    1.
    \int\frac{-x}{(x + 1) - \sqrt{x + 1}} dx

    2.
    \int^2_1(x - 1)\sqrt{2 - x} dx

    I am stumped on both of these problems.
    On #1, I try u = x + 1, but wind up with
    \int\frac{-u + 1}{u - u^\frac{1}{2}} du
    which only seems more unsolvable than it was before

    On #2, I try u = 2 - x, but then wind up getting 1 for the lower limit and 0 for the upper limit which makes no sense.

    Help, please?
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  2. #2
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    Quote Originally Posted by seuzy13 View Post
    1.

    \int\frac{-u + 1}{u - u^\frac{1}{2}} du
    which only seems more unsolvable than it was before
    This might help.

    \int\frac{-u }{u - u^\frac{1}{2}} du+\int\frac{ 1}{u - u^\frac{1}{2}} du

    Now factor out u and cancel in the first integral.
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  3. #3
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    Quote Originally Posted by seuzy13 View Post
    On #2, I try u = 2 - x, but then wind up getting 1 for the lower limit and 0 for the upper limit which makes no sense.
    That does make sense. Consider \int_a^b f=g(b)-g(a)=-\big(g(a)-g(b)\big)=-\int_b^a f, so if you flip the bounds, then the integral changes its sign.
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  4. #4
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    Hmm...but now I'm winding up with:
    \int\frac{-1}{1 - u^\frac{-1}{2}} + \frac{1}{u} - \frac{1}{u^\frac{1}{2}} du
    In which I the first term still seems unsolvable...?

    EDIT--
    Oh and thanks for the clarification on #2. I had forgotten about that rule.
    Last edited by seuzy13; January 23rd 2010 at 01:58 PM.
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  5. #5
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    \int{\frac{-x}{x+1-\sqrt{x+1}}}dx=-\int{\frac{x}{\sqrt{x+1}^2-\sqrt{x+1}}}dx=-\int{\frac{u^2-1}{u^2-u}}2udu

    where u=\sqrt{x+1}

    so x=u^2-1

    giving

    -\int{\frac{(u+1)(u-1)}{u(u-1)}}2udu=-2\int{(u+1)}du=-2(\frac{u^2}{2}+u)+C

    =-2\ \frac{x+1}{2}-2\sqrt{x+1}+C



    sorry about the typo, seuzy
    Last edited by Archie Meade; January 23rd 2010 at 05:14 PM. Reason: typo
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  6. #6
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    Double check your substitution, Archie Meade!


    I think this is the easiest method:


    Let u=x+1 \implies x=u-1


    <br />
\int \frac{-x}{x+1-\sqrt{x+1}} dx<br />
=\int \frac{-(u-1)}{u-u^{1/2}}du<br />
=\int \frac{-(u^{1/2}+1)(u^{1/2}-1)}{u^{1/2}(u^{1/2}-1)}du<br />

    <br />
=\int \frac{-(u^{1/2}+1)}{u^{1/2}}du<br />
=\int \left(-1 - u^{-1/2}\right) du<br />
= -u - 2u^{1/2} = ...
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