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Math Help - Improper integral help

  1. #1
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    Improper integral help

    So this was on my test today and I couldnt figure it out.

    int[ (x+2) / (x+5) ]

    obviously part of the answer was ln|x+5|, but i couldnt figure out what to do with the (x+2)

    thanks in advance!
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  2. #2
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    Quote Originally Posted by sj9110 View Post
    So this was on my test today and I couldnt figure it out.

    int[ (x+2) / (x+5) ]

    obviously part of the answer was ln|x+5|, but i couldnt figure out what to do with the (x+2)

    thanks in advance!
    by long division we get

    \frac{x+2}{x+5}=1-\frac{3}{x+5}

    This can now be integrated to give

    x-3\ln(x+5)+c
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    Quote Originally Posted by sj9110 View Post
    So this was on my test today and I couldnt figure it out.

    int[ (x+2) / (x+5) ]

    obviously part of the answer was ln|x+5|, but i couldnt figure out what to do with the (x+2)
    \int\frac{x+2}{x+5}\,dx

    =\int\frac{x+2\color{red}+3-3}{x+5}\,dx

    =\int\frac{x+5-3}{x+5}\,dx

    =\int\left(\frac{x+5}{x+5}-\frac3{x+5}\right)dx

    =\int\left(1-\frac3{x+5}\right)dx

    =\int dx-3\int\frac1{x+5}\,dx

    =x-3\ln|x+5|+C
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