Am I right to assume that WHENEVER an expression containing a natural logarithm is produced from integrating an expression, the expression always contain the absolute value symbol? The reason I'm asking this is because my book sometimes confuse me by alternating between using the absolute value symbol and forgoing it when producing a natural logarithmic expression from integration.
Also, why do we even use the absolute value symbol for natural logarithms? We know that ln[abs(x)] = ln x.
I've another question. Let's use the following example -
INTEG: (4x^2 - 2x + 2)/[(x^2 + 1)(x-1)] dx
After using partial fractions, the answer is, according to my book, ln[abs(x^2 + 1)] + 2 ln [abs(x-1)] + C
Now, obviously, the first ln expression cannot be negative so why is there an absolute value symbol in it? Is it because of the other ln expression? I'd appreciate some clarification. Thanks!