# Integral of quotient with natural log

• Jun 25th 2012, 04:56 PM
Ragnarok
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?
• Jun 25th 2012, 05:18 PM
Reckoner
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 $x > 0.$
• Jun 25th 2012, 06:17 PM
Ragnarok
Re: Integral of quotient with natural log
Thanks so much!