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Math Help - Evaluate the following integral

  1. #1
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    Exclamation Evaluate the following integral

    Question :
    \frac{dx}{(1+x)\sqrt{x}}

    Please answer in steps
    Thank you
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  3. #3
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    =(1/1+x)(x^-0.5)
    =arctan(?) 1/1+?^2=1/1+x
    ?=x^0.5
    But Arctan (x^0.5)=1/[2(x+1)(x^0.5)]
    Therefor we need 2arctan(x^0.5).
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  4. #4
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    Hello, mj.alawami!

    \frac{dx}{(1+x)\sqrt{x}}

    Let: . u \,=\,\sqrt{x} \quad\Rightarrow\quad x \,=\,u^2\quad\Rightarrow\quad dx \,=\,2u\,du


    Substitute: . \int\frac{2u\,du}{(1+u^2)\,u} \;=\;2\!\!\int\!\frac{du}{1+u^2} \;=\; 2\arctan u + C


    Back-substitute: . 2\arctan\left(\sqrt{x}\right) + C

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