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Math Help - Substitution/Integration by Parts

  1. #1
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    Substitution/Integration by Parts

    Hi, I'm having a little trouble with a substitution/integration by parts question...

    find the integral of: ln(sqrt(x) +1)

    the book lists the first step as substituting y = sqrt(x) +1

    from there i tried integration by parts using u=lny, v'=1 but it didn't quite work out

    the solution in meant to be: sqrt(x) - x/2 + (x-1)ln(1+sqrt(x))

    any suggestions?

    thank you!
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  2. #2
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    Quote Originally Posted by strgrl View Post
    find the integral of: ln(sqrt(x) +1)

    the book lists the first step as substituting y = sqrt(x) +1
    Now contemplate its derivative dy=\frac1{2\sqrt x}\,dx.

    This means that dy = \frac{1}<br />
{{2(y - 1)}}\,dx. So

    \int {\ln \left( {1 + \sqrt x } \right)\,dx}  = 2\int {(y - 1)\ln y\,dy} .

    From here it's quite straightforward, so just split the original integral into two pieces, both of them are easy to tackle.

    (If x was bounded on [0,1], a definite integral, I could kill this with a double integration trick.)
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    thank you!
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