Integral definition doubt

[JPJ]

Hello there,

I'm a beginner to the study of Calculus, and I'm wondering why does integral dx = x + C, but, for example, integral dx/x = ln(x) + C ?
To be more precise, my doubt is what happened to the dx in the integrals with other elements involved? It seems like only 1/x was integrated.

Thanks in advance.

 

[pka]

It might interested to know that there is a whole group of analysis who do not use the \(\displaystyle dx\) when writing an integral.
Instead of \(\displaystyle \int_0^2 {(x^2 + 2x)dx} \) they simply use \(\displaystyle \int_0^2 {x^2 + 2x} \).

If you realize that \(\displaystyle \int {dx} \) is really \(\displaystyle \int {1 dx} \) then it makes more sense.
Some people tell students that the \(\displaystyle dx\) tells us the variable.

So in \(\displaystyle \int {\left( {a^2 - 3x^2 } \right)dx} \) we know that the antiderivative is \(\displaystyle a^2x-x^3+C\).

 

[JPJ]

I see. But if I have integral 1/x * dx, what forbids me to solve it using the chain rule?
It would go like: integral 1/x * dx = x/x - integral [d(1/x)]/x *x*dx <=> integral 1/x * dx = 1 - integral (-1)*(x-² )*x*dx <=> integral 1/x * dx = 1 + integral 1/x * dx, always ending up with repetitions of the original integral 1/x * dx .
I got a misconception somewhere, because I'm sure that is incorrect.

Btw, u type the equations using a latex editor or...?

 

[pka]

I see. But if I have integral 1/x * dx, what forbids me to solve it using the chain rule?
It would go like: integral 1/x * dx = x/x - integral [d(1/x)]/x *x*dx <=> integral 1/x * dx = 1 - integral (-1)*(x-² )*x*dx <=> integral 1/x * dx = 1 + integral 1/x * dx, always ending up with repetitions of the original integral 1/x * dx .
I got a misconception somewhere, because I'm sure that is incorrect.
Btw, u type the equations using a latex editor or...?

Well \(\displaystyle \int {\frac{1}{x}dx} \) is a very special integral and is \(\displaystyle \ln(x)+c\).