# Integral of quotient with natural log

• Jun 25th 2012, 03:56 PM
Ragnarok
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?
• Jun 25th 2012, 04:18 PM
Reckoner
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 $\displaystyle x > 0.$
• Jun 25th 2012, 05:17 PM
Ragnarok
Re: Integral of quotient with natural log
Thanks so much!