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Math Help - Substitution Method Help

  1. #1
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    Substitution Method Help

    The substitution u=2x+7 transforms the "x"-integral
    \displaystyle \int \frac{4 x + 3}{\sqrt{2 x + 7}} \ dx
    into the "u"-integral

    ok, so I can rewrite this as:

    \displaystyle \int (4 x + 3)(2 x + 7)^{\frac{-1}{2}} \ dx
    u=2x+7
    du=2dx
    I can't figure out how to force this to equal 4x+3

    All I can do is
    2\int u^{\frac{-1}{2}}*du

    which will give me my 4, but I'm still missing the x+3 part!

    What am I missing?
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  2. #2
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    Quote Originally Posted by Vamz View Post
    The substitution u=2x+7 transforms the "x"-integral
    \displaystyle \int \frac{4 x + 3}{\sqrt{2 x + 7}} \ dx
    into the "u"-integral

    ok, so I can rewrite this as:

    \displaystyle \int (4 x + 3)(2 x + 7)^{\frac{-1}{2}} \ dx
    u=2x+7
    du=2dx
    I can't figure out how to force this to equal 4x+3

    All I can do is
    2\int u^{\frac{-1}{2}}*du

    which will give me my 4, but I'm still missing the x+3 part!

    What am I missing?
    Dear Vamz,

    \displaystyle \int \frac{4 x + 3}{\sqrt{2 x + 7}} \ dx

    u=2x+7\Rightarrow{dx=\frac{du}{2}}

    Also, x=\frac{u-7}{2}

    Substituting these in the above integration;

    \displaystyle \int \frac{4 \left(\frac{u-7}{2}\right) + 3}{2\sqrt{u}} \ du

    \displaystyle\frac{1}{2}\int \frac{2u-11}{\sqrt{u}}du

    Hope you can continue.
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