Page:Calculus Made Easy.pdf/213

Thus, if ${\dfrac {dy}{dx}}=4x^{2}$, the reverse process gives us $y={\tfrac {4}{3}}x^{3}$.

But this is incomplete. For we must remember that if we had started with

$y=ax^{n}+C$,

where $C$ is any constant quantity whatever, we should equally have found

${\frac {dy}{dx}}=anx^{n-1}$.

So, therefore, when we reverse the process we must always remember to add on this undetermined constant, even if we do not yet know what its value will be.

This process, the reverse of differentiating, is called integrating; for it consists in finding the value of the whole quantity $y$ when you are given only an expression for $dy$ or for ${\dfrac {dy}{dx}}$. Hitherto we have as much as possible kept $dy$ and $dx$ together as a differential coefficient: henceforth we shall more often have to separate them.

If we begin with a simple case,

${\frac {dy}{dx}}=x^{2}$.

We may write this, if we like, as

$dy=x^{2}\,dx$.

Now this is a “differential equation” which informs us that an element of $y$ is equal to the corresponding element of $x$ multiplied by $x^{2}$. Now, what we want