# Is it wrong to use absolute value brackets instead of brackets if the value inside it is always positive?

We know that $$∫{1\over x} dx$$ $$=$$ $$ln \lvert x\rvert$$ $$+$$ $$c$$

If the $$x$$ within the natural logarithm is always positive, for example $$x$$ $$=$$ $$3x^2$$ $$+$$ $$5$$, should $$∫{x\over 3x^2+5} dx$$ be expressed as

$$\int{x\over 3x^2+5} dx ={1\over6}\ln \lvert 3x^2+5\rvert+ c$$

OR

$$\int{x\over 3x^2+5} dx={1\over6}\ln (3x^2+5)$$ $$+$$ $$c$$

Would it be wrong to use absolute value brackets instead of brackets if the value inside is always positive? Is it necessary to take into account the possibility of $$x$$ being a complex number / an imaginary number and thus making $$\ln (3x^2+5)$$ negative?

Which expression is more accurate?

• Logs do not work trivially with complex numbers, especially when it comes to integrals (and even more so for changes of variable). I would assume real-valued $x$ in which case you can feel free to keep or remove the absolute value bars. The formula with absolute values is a real-valued phenomenon anyway. (There is an extension for complex numbers.) Commented Jul 31, 2022 at 16:16
• Nothing that is true is "wrong". It may be obfuscating, redundant, inefficient, confusing and unnecessary. But it isn't wrong. Commented Jul 31, 2022 at 16:19
• Check this out for similar discussion: math.stackexchange.com/questions/1154321/… Commented Jul 31, 2022 at 16:22
• Commented Jul 31, 2022 at 16:28

Usually, the convention for a logarithm as the answer to an indefinite integral is $$\ln |z|+C$$, but if $$|z|$$ is positive, $$\ln (z) + C$$ is appropriate. In general for indefinite integrals, if it assumed that any variable will be positive, the absolute value bars are not required and can be dropped - but for logarithms, I would still use parentheses if the expression is part of the logarithm.
In your case, either $$\int \dfrac {x \ dx}{3x^2 +5} = \frac {1}{6} \ln |3x^2 + 5|+C$$ or $$\int \dfrac {x \ dx}{3x^2 +5} = \frac {1}{6} \ln (3x^2 + 5)+C$$ is perfectly fine as an answer as $$3x^2 + 5$$ is positive for all $$x$$.
It's when you come to definite integrals where the absolute value bars may be necessary. If $$x \leq 0$$, $$\ln x$$ is undefined in the reals; but if we get a value inside the absolute value bars that evaluates to a positive real number, the absolute value bars are not needed.