If $dF=f(x)dx$, should I write $\int_a^b f(x)dx$ as $\int_a^bdF$? or as $\int_{F(a)}^{F(b)}dF$? What is a proper way to change the differential of an integral? For example suppose we have the following integral:
$$\int_1^2 2x dx$$
which equals 3.
But we know that $2x dx = d(x^2)$. Should I write:
$$\int_1^2 2x dx= \int_1^2 d(x^2)$$
or
$$\int_1^2 2x\cdot dx= \int_1^4 d(x^2)$$
This is how I would interpret and calculate the integrals. With the first notation:
$$\int_1^2 d(x^2)=x^2\Big|_1^2=3$$
whereas with the second notation:
$$\int_1^4 d(x^2)=x^2\Big|_1^4=16-1=15$$
In essence my problem is how I should interpret the limits and the differential inside the integral.
I picked just this example but I can generalize my confusion to any integral with arbritrary limits:
$$\int_a^b f(x)dx = \int_a^bdF$$
or
$$\int_a^b f(x)dx = \int_{F(a)}^{F(b)}dF$$
where $dF=f(x)dx$ that is $f(x)$ is the derivative of $F(x)$.

You may write
$$\int_a^b f(x)dx = \int_{x=a}^{x=b} dF$$
I think this would be more accurate
