Thread: Integral of quotient with natural log

1. 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?

2. 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 $x > 0.$

3. Re: Integral of quotient with natural log

Thanks so much!