1. the question is (integral) x log x dx

i did integration by parts, and got it right with one exceptino....for the first term, i got 1/2 x^2 log x
however, in the answers it says the x after the log should be in absolute value. why is this (or how do i show this in my work), i set my u to "log x" so i dont understand why it would change....

any help as always is appreciated!

would it be sufficient if i just threw the absolute value in the first step.. is this understood to be true without explanation?

2. Originally Posted by twostep08
the question is (integral) x log x dx

i did integration by parts, and got it right with one exceptino....for the first term, i got 1/2 x^2 log x
however, in the answers it says the x after the log should be in absolute value. why is this (or how do i show this in my work), i set my u to "log x" so i dont understand why it would change....

any help as always is appreciated!

would it be sufficient if i just threw the absolute value in the first step.. is this understood to be true without explanation?
$\int x \log x dx$

let $u=\log x , dv= x \Rightarrow du=\frac{1}{x} , v=\frac{x^2}{2}$

$\int x\log x dx =\frac{x^2}{2} \cdot \log x - \int \frac{x^2}{2x} dx$

$\int x\log x dx = \frac{\log x \cdot x^2}{2} - \frac{x^2}{2\cdot 2}+C$

log(x) is defined for x positive values just (i,e x>0), for example log(-4) is not a real number