Integral of quotient with natural log

**Ragnarok** · June 25th 2012, 03:56 PM

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?

**Reckoner** · June 25th 2012, 04:18 PM

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

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

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

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

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

The absolute value bars are necessary unless we assume that $x > 0.$

**Ragnarok** · June 25th 2012, 05:17 PM

Thanks so much!

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