Page:Calculus Made Easy.pdf/214

is the integral; therefore, write down with the proper symbol the instructions to integrate both sides, thus:

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

[Note as to reading integrals: the above would be read thus:

“Integral dee-wy equals integral eks-squared dee-eks.”]

We haven’t yet integrated: we have only written down instructions to integrate–if we can. Let us try. Plenty of other fools can do it–why not we also? The left-hand side is simplicity itself. The sum of all the bits of $y$ is the same thing as $y$ itself. So we may at once put:

$y=\int x^{2}\,dx$.

But when we come to the right-hand side of the equation we must remember that what we have got to sum up together is not all the $dx$’s, but all such terms as $x^{2}dx$; and this will not be the same as $x^{2}\int dx$, because $x^{2}$ is not a constant. For some of the $dx$’s will be multiplied by big values of $x^{2}$, and some will be multiplied by small values of $x^{2}$, according to what $x$ happens to be. So we must bethink ourselves as to what we know about this process of integration being the reverse of differentiation. Now, our rule for this reversed process–see p. 191 ante–when dealing with $x^{n}$ is “increase the power by one, and divide by the same number as this increased power.”