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Math Help - Integration using logs

  1. #1
    Member Awsom Guy's Avatar
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    Integration using logs

    Hello I am having trouble with this question.

    INtegrate:
    f(2x+5)/(x+4) dx

    My attempt:
    1/2x+5f 1/x+4 dx
    2x+5 Ln (x+4) + c
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  2. #2
    Eater of Worlds
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    \int\frac{2x+5}{x+4}dx=2x-3ln(x+4)+C

    You're close. But there should be a minus where the plus is, and a 3 instead of a 5 in front of the ln.
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  3. #3
    Member Awsom Guy's Avatar
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    But how do you get the -3. That is the part I cannot do.
    By the way thanks for the help and nice avatar thing.
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  4. #4
    Eater of Worlds
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    If we expand, we get

    \frac{2x+5}{x+4}=2-\frac{3}{x+4}

    Now, integrate that and you can see why.
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  5. #5
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    Alternatively,

    \int{\frac{2x+5}{x+4}}dx=\int{\frac{x+4+x+1}{x+4}}  dx=\int{\left(1+\frac{x+1}{x+4}\right)}dx

    u=x+4

    x+1=u-3

    \int{\left(1+\frac{u-3}{u}\right)}du=\int{\left(1+1-3u^{-1}\right)}du=\int{\left(2-3u^{-1}\right)}du

    =2u-3ln|u|+C=2(x+4)-3ln|x+4|+C
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