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Math Help - Evaluating indefinite Integral

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    Evaluating indefinite Integral

    Could someone please work me out in these problem, I did a substitution on Ln(x) but ended up with (u^2+1)/(u+1)...
    here is the question find the indefinite integral of ((ln x)^2+1)/(xln(x)+3)....
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    Re: Evaluating indefinite Integral

    Are you sure it's not \displaystyle \begin{align*} \int{\frac{\left( \ln{x} \right)^2 + 1}{x \left( \ln{x} + 3 \right)}\,dx} \end{align*} instead of \displaystyle \begin{align*} \int{\frac{\left( \ln{x} \right)^2 + 1}{x\ln{x} + 3}\,dx} \end{align*}?
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    Re: Evaluating indefinite Integral

    Oh I'm very sorry, it is actually (x ln x +3x)..... the 3 has 'x' with it..
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    Re: Evaluating indefinite Integral

    So it is what I thought.

    I would write it as \displaystyle \begin{align*} \int{\frac{\left( \ln{x} \right)^2 + 1}{\ln{x} + 3}\cdot \frac{1}{x}\,dx} \end{align*}.

    Now this is just ONE possible substitution. I'm choosing it because I know that it will make the denominator the easiest to work with.

    Let \displaystyle \begin{align*} u = \ln{x} + 3 \implies du = \frac{1}{x} \, dx \end{align*}. Then your integral becomes

    \displaystyle \begin{align*} \int{\frac{\left( \ln{x} \right)^2 + 1}{\ln{x} + 3}\cdot \frac{1}{x} \,dx} &= \int{\frac{\left( u - 3 \right)^2 + 1}{u}\,du} \\ &= \int{\frac{u^2 - 6u + 9 + 1}{u}\,du} \\ &= \int{\frac{u^2 - 6u + 10}{u}\,du} \\ &= \int{u - 6 + \frac{10}{u}\,du} \\ &= \frac{u^2}{2} - 6u + 10\ln{|u|} + C \\ &= \frac{\left( \ln{x} + 3 \right)^2}{2} - 6\left( \ln{x} + 3 \right) + 10\ln{\left| \ln{x} + 3 \right|} + C \end{align*}
    Thanks from MarkFL and kspkido
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    Re: Evaluating indefinite Integral

    daym... I got the u^2-6u+10/u... but at that point I didn't know what to do.. so I tried to substitute other variables.. stupid me... Thank you very much...
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