Question

Evaluate: $\int |x|\ln |x^2|dx$

• A. $-\frac{x^2}{2}\ln |x^2|+\frac{x^2}{2}+c,x>0$
• B. $\frac{1}{2}x|x|\ln |x^2|-\frac{1}{2}x|x|+c,x\ne 0$
• C. $-\frac{x^2}{2}\ln |x^2|+\frac{x^2}{2}+c,x<0$
• D. $\frac{1}{2}x|x|\ln |x^2|-\frac{1}{2}x|x|+c$

Answer 1

Answer: (B), (C)

$\frac{1}{2}x|x|\ln |x^2|-\frac{1}{2}x|x|+c,x\ne 0$
$-\frac{x^2}{2}\ln |x^2|+\frac{x^2}{2}+c,x<0$

$\int |x|\ln |x^2|dx$

for $x<0$

$\int -x|\ln |x^2|dx$

$\int -|\ln |x^2|\frac{d\left(x^2\right)}{2}$ Since $xdx=\frac{d\left(x^2\right)}{2}$

$=-\frac{1}{2}[\ln |x^2|.x^2-\int x^2.\frac{1}{x^2}2xdx]$

$=-\frac{1}{2}[\ln |x^2|.x^2-\frac{{2x}^2}{2}+c]$

$=-\frac{1}{2}\ln |x^2|.x^2+\frac{x^2}{2}+c^{\prime }$

for $x>0$

$\int |x|\ln |x^2|dx$

$\int |\ln |x^2|\frac{d\left(x^2\right)}{2}$

$=\frac{1}{2}[\ln |x^2|.x^2-\int x^2.\frac{1}{x^2}2xdx]$

$=\frac{1}{2}[\ln |x^2|.x^2-\frac{{2x}^2}{2}+c]$

$=\frac{1}{2}\ln |x^2|.x^2-\frac{x^2}{2}+c$

for $x>0\to \frac{1}{2}\ln |x^2|.x^2-\frac{x^2}{2}+c$

$x<0\to -\frac{1}{2}\ln |x^2|.x^2+\frac{x^2}{2}+c^{\prime }$

$x\ne 0\to \frac{1}{2}\ln |x^2|.x|x|-\frac{x|x|}{2}+c$

Answer B,C