Page:Calculus Made Easy.pdf/211

CHAPTER XVIII.

INTEGRATING AS THE REVERSE OF DIFFERENTIATING.

Differentiating is the process by which when $y$ is given us (as a function of $x$), we can find ${\frac {dy}{dx}}$.

Like every other mathematical operation, the process of differentiation may be reversed; thus, if differentiating $y=x^{4}$ gives us ${\frac {dy}{dx}}=4x^{3}$; if one begins with ${\frac {dy}{dx}}=4x^{3}$ one would say that reversing the process would yield $y=x^{4}$. But here comes in a curious point. We should get ${\frac {dy}{dx}}=4x^{3}$ if we had begun with any of the following: $x^{4}$, or $x^{4}+a$, or $x^{4}+c$, or $x^{4}$ with any added constant. So it is clear that in working backwards from ${\frac {dy}{dx}}$ to $y$, one must make provision for the possibility of there being an added constant, the value of which will be undetermined