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Math Help - Integral by parts

  1. #1
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    Integral by parts

    Integral from 9 to 16 lny/sqrt(y)

    Here's my work and on the parts bolded I asked for help from someone else but Im not sure WHY he did those steps:

    ∫ ln(y)⋅y^(−) dy =
    2⋅y^()⋅ln(y) − ∫ 2⋅y^()⋅(1/y) dy
    2⋅y^()⋅ln(y) − 2⋅∫ y^(−) dy
    2⋅y^()⋅ln(y) − 2⋅(2)⋅y^() + C
    2⋅y^()⋅ln(y) − 4⋅y^() + C
    2⋅y^()⋅[ ln(y) − 2 ] + C
    2⋅√(y)⋅[ ln(y) − 2 ] + C

    Im not sure why and how the 4y1/2 got eliminated/broken up from the problem. Can anyone explain why these two parts happen? I can handle the rest of the problem from here as you just have to plug in the two intervals and subtract them from each other.
    Thanks!
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  2. #2
    Senior Member MacstersUndead's Avatar
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    Re: Integral by parts

    All he did was factor out a 2y^{\frac{1}{2}} from the first two terms and then recall y^{\frac{1}{2}} = \sqrt{y}. It was for simplification only
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    Re: Integral by parts

    Let's see, I'm just gonna go from the first bold line to the line above it:
     2y^{1/2}\cdot (\ln(y)-2)+C=\ln(y)\cdot  2y^{1/2}-2\cdot2y^{1/2} +C = 2y^{1/2}\cdot \ln(y)-4y^{1/2} +C
    and the last bold line is just because by definition  y^{1/2}=\sqrt{y}.
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  4. #4
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    Re: Integral by parts

    OH! I see now. All he did was factor...wow. I appreciate both of your help!
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