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Thread: 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:

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

    Letting $\displaystyle u = lnx^2$ I get

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

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

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

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

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

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

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

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

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

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

    Thanks so much!
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