Answer
(a) is incorrect and (b) is correct.
Work Step by Step
Here, we have
$x \int \ln |x| dx=x(\ln |x|+c$
or, $x \int \ln |x| dx=x\ln |x|+cx$
Thus, it can be seen that solution (a) is incorrect because here the arbitrary constant depends on the values of $c$ but solution (b) is correct because it is the same as the solution.
Hence, (a) is incorrect and (b) is correct.
