The logarithm is defined as: $$ \ln x = \int_1^x \frac1{t} dt $$ Hence I am often told that for indefinite integrals, since $\frac1{x}$ is defined over $\mathbb{R} \setminus \{0\}$ (various sources seem to disagree about this), one should write: $$ \int \frac1{x}dx = \ln |x|+C $$ How is this justified?
(PS I think I might be duplicating someone else, if I am please link)
On
It is said that $\int \frac{1}{x}$d$x=\ln |x| +c$ because we must account for when $x<0$ as well as when $x>0$. I believe the justification is derived from Euler's identity:
$e^{i \pi}+1=0 \implies e^{i \pi}= -1 \implies i\pi=\ln(-1)$
When $x<0$, then we have the indefinite integral giving $\ln (-x)+c_1=\ln (x)+ c +i\pi=\ln (x) +c_2$
So they differ by $i\pi$.
One writes $\displaystyle \int \frac1x dx=\ln|x|$, because you should take into account cases where $x<0$.
In the definition $\displaystyle \ln x = \int_1^x \frac1{t} dt$, only the values for $x>0$ are taken into account.