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Math Help - simple integration problem

  1. #1
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    simple integration problem

     \int_4^9 \frac{1}{(2x)(1+ \sqrt x)}
    please help
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  2. #2
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    Re: simple integration problem

    Quote Originally Posted by earthboy View Post
     \int_4^9 \frac{1}{(2x)(1+ \sqrt x)}
    please help
    Assuming you mean \displaystyle \begin{align*} \int_4^9{\frac{1}{2x\left(1 + \sqrt{x}\right)}\,dx} &= \int_4^9{\frac{1}{2\sqrt{x}\sqrt{x}\left(1 + \sqrt{x}\right)}\,dx} \end{align*}

    Now make the substitution \displaystyle \begin{align*} u = \sqrt{x} \implies du = \frac{1}{2\sqrt{x}}\,dx \end{align*} and also note that this means \displaystyle \begin{align*} 1 + \sqrt{x} = 1 + u \end{align*} and the limits will change to 2 and 3, and the integral becomes

    \displaystyle \begin{align*} \int_4^9{\frac{1}{2\sqrt{x}\sqrt{x}\left(1 + \sqrt{x}\right)}\,dx} = \int_2^3{\frac{1}{u\left(1 + u\right)}\,du} \end{align*}

    which you can solve using partial fractions.
    Thanks from earthboy
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  3. #3
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    Re: simple integration problem

    Substitute u=x^1/2
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