Integration

Let \(y=F(x)\), then the differential is defined as

$$ dy=dF=F^\prime (x)dx $$

where \(dy\) is the infinitesimal difference in \(y\), and \(dx\) is the infinitesimal difference in \(x\). \(dy\) or \(dF\) can be used interchangeably.

The indefinite integral = antiderivative = inverse differential

Why to use differential instead of derivative?

We know from the previous course that the derivative of a function is the limit of the quotient of the difference in y to the difference in x as the difference in x approaches zero.

$$ f^\prime (x) = \frac{df}{dx} = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0} $$

We can think of the derivative as a function that does something with another function. Once we understood how the derivative works, we want to know the inverse of it in order to undo what the derivative does. We call this function antiderivative (or indefinite integral). The problem with it is that it does not take the derivative as an input, instead it takes the differential as an input, producing the original function in the output. Why do we call it antiderivative instead of antidifferential then? Because it reverses the job done by the derivative! But it is easy to make a differential from a derivative: all we need to do is to multiply the derivative by dx, which is the infinitesimal change in x from the formula of the derivative.

How to live with this notation? Personally, I think of antiderivative as a very finicky function. It does what we want it to do, but needs some help: it should be multiplied by the infinitesimal change in x, otherwise it does not work. As an example illustrating why it is so, consider

$$ f(x)=x^2 $$

$$ f^\prime (x)=\frac{df}{dx}=\frac{dx^2}{dx}=2x\quad=>\quad dx^2=2xdx\quad=> $$

$$ f(x)=\int df=\int dx^2=\int 2xdx\quad=>\quad f(x)=x^2 $$