`int |x|ln|x|dx` equals `(x ne 0)`

Options

A. `(x^(2))/(2)ln|x|-(x^(2))/(4)+C`
B. `(1)/(2)x|x|lnx+(1)/(4)x|x|+C`
C. `-(x^(2))/(2)ln|x|+(x^(2))/(4)+C`
D. `(1)/(2)x|x|ln|x|-(1)/(4)x|x|+C`

Answer

D

Explanation

Case I If x gt 0, then `|x|=x`
On applying integration by parts, we get
`therefore" "=intx ln x dx=ln x.(x^(2))/(2)-int(1)/(x).(x^(2))/(2)dx`
`=(x^(2))/(2).lnx-(x^(2))/(4)+C`
Case II If x lt 0, then `|x|=-x`
`therefore" "int |x|ln |x| dx = - int x ln (-x) dx`
`-{ln(-x).(x^(2))/(2)-(x^(2))/(4)}+C`
`-(x^(2))/(2)ln|x|+(x^(2))/(4)+C`
Combining both cases, we get
`(1)/(2)x|x|ln|x|-(1)/(4)x|x|+C`