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Math Help - Integral of (z+2)Sqrt(1-z)dz

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    Integral of (z+2)Sqrt(1-z)dz

    Integral of (z+2)Sqrt(1-z)dz-capture.jpgI need to integrate this using u-substitution, integration by parts, and this integral table. I'm having a difficult time getting started. U substitution doesn't seem viable as I can't identify the derivative of any U within the integral. I'm assuming I need to use integration by parts, but internal integration and derivation only seems to provide a more difficult integral. The integral table doesn't seem to have any forms that are relevant to this problem.
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    Re: Integral of (z+2)Sqrt(1-z)dz

    Quote Originally Posted by bikerboy2442 View Post
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ID:	26683I need to integrate this using u-substitution, integration by parts, and this integral table. I'm having a difficult time getting started. U substitution doesn't seem viable as I can't identify the derivative of any U within the integral. I'm assuming I need to use integration by parts, but internal integration and derivation only seems to provide a more difficult integral. The integral table doesn't seem to have any forms that are relevant to this problem.
    First, rewrite this as \displaystyle \begin{align*} -\int{-(z + 2)\sqrt{1 - z}\,dz} \end{align*} and then substitute \displaystyle \begin{align*} u = 1 - z \implies du = -dz \end{align*}. Also note that if \displaystyle \begin{align*} u = 1 - z \end{align*}, then \displaystyle \begin{align*} z + 2 = 3 - u \end{align*}. The integral becomes

    \displaystyle \begin{align*} -\int{-(z + 2)\sqrt{1 - z}\,dz} &= -\int{ \left(3 - u \right) \sqrt{u}\,du} \\ &= -\int{ \left(3 - u \right) u^{\frac{1}{2}}\,du} \\ &= -\int{3u^{\frac{1}{2}} - u^{\frac{3}{2}}\,du} \end{align*}

    You should be able to go from here. Note there is no need for Integration by Parts.
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