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Math Help - Integral of quotient with natural log

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    Integral of quotient with natural log

    Hello, I am having trouble with another Calculus 2 problem:

    \int \frac{lnx^2}{x}dx

    Letting u = lnx^2 I get

    \frac{1}{2} \int u \,du

    = \frac{1}{2}\left[ \frac{u^2}{2} \right] + C

    = \frac{(lnx^2)^2}{4} + C

    However, the textbook gives (lnx)^2 + C. Is this somehow another form, or what am I doing wrong?
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    Re: Integral of quotient with natural log

    \int\frac{\ln x^2}x\,dx

    =\frac14\left(\ln x^2\right)^2 + C

    =\frac1{2^2}\left(\ln x^2\right)^2 + C

    =\left(\frac12\ln x^2\right)^2 + C

    =\left(\ln|x|\right)^2 + C

    The absolute value bars are necessary unless we assume that x > 0.
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    Re: Integral of quotient with natural log

    Thanks so much!
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