# Can somebody provide an explanation to the formula of a one elementary integral?

Here is the formula:

$$\int{\frac{dx}{x}} = \ln{|x|} + C$$

In my textbook it is given without proof, so I have a little confusion here. From the definition of integral this equality must be true:

$$(\ln|x| + C) = \frac{1}{x}$$

But I failed to derive it. Cause according to the rule of differentiation of a complex funtion $(f_1(f_2(x))) = f_1(f_2(x))*f_2(x)$ (if I understand it right) the derivative of $\ln|x| + C$ is:

$$(\ln|x| + C) = (\ln|x|) + C = \frac{1}{|x|} * |x| + 0$$

And this is confirmed by the Wolfram Mathematica:

So this is obviously not $\frac{1}{x}$. Can somebody provide an explanation for this problem?

My appreciation.

• If $x>0$ then $|x|'=1$ and $|x|'/|x|=1/x$. If $x<0$ then $|x|'=-1$ and $|x|'/|x|=-1/(-x)=1/x$. May 8, 2013 at 11:57
• @Artem, thank you very much!
– d.k
May 8, 2013 at 12:01

$|x|'=Sgn(x)$ and $\frac{\operatorname{Sgn}(x)}{|x|}=\frac{1}{x}$, where $\operatorname{Sgn(x)}=1$ if $x\geq 0$, and $\operatorname{Sgn(x)}=-1$ if $x< 0$