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Math Help - Evaluating integrals with radicals and fractions

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    Evaluating integrals with radicals and fractions

    Two integration problems are giving me a lot of trouble.


    (2x + 2)/(x^2 + 2x)^(1/2)


    and


    dx/(x^1/2)(1 - x^1/2)^4



    I can't get the substitution technique to work out on these two for some reason.
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    \int{\frac{2x+2}{(x^2 + 2x)^{\frac{1}{2}}}\,dx} = \int{(2x+2)(x^2+2x)^{-\frac{1}{2}}\,dx}.

    Let u = x^2+2x so that du = (2x+2)dx and the integral becomes

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

     = \frac{u^{\frac{1}{2}}}{\frac{1}{2}}+C

     = 2u^{\frac{1}{2}}+C

     = 2(x^2+2x)^{\frac{1}{2}}+C.
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    \int{\frac{dx}{x^{\frac{1}{2}}(1-x^{\frac{1}{2}})^4}} = \int{x^{-\frac{1}{2}}(1-x^{\frac{1}{2}})^{-4}\,dx}

     = -2\int{-\frac{1}{2}x^{-\frac{1}{2}}(1-x^{\frac{1}{2}})^{-4}\,dx}.

    Let u=1-x^{\frac{1}{2}} so that du = -\frac{1}{2}x^{-\frac{1}{2}}\,dx and the integral becomes

    -2\int{u^{-4}\,du}

     = -2\left(\frac{u^{-3}}{-3}\right) + C

     = \frac{2}{3u^3} + C

     = \frac{2}{3(1-x^{\frac{1}{2}})^3} + C.
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