Page:Calculus Made Easy.pdf/212

until ascertained in some other way. So, if differentiating $x^{n}$ yields $nx^{n-1}$, going backwards from ${\dfrac {dy}{dx}}=nx^{n-1}$ will give us $y=xn+C$; where $C$ stands for the yet undetermined possible constant.

Clearly, in dealing with powers of $x$, the rule for working backwards will be: Increase the power by $1$, then divide by that increased power, and add the undetermined constant.

So, in the case where

${\dfrac {dy}{dx}}=x^{n}$,

working backwards, we get

$y={\frac {1}{n+1}}x^{n+1}+C$.

If differentiating the equation $y=ax^{n}$ gives us

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

it is a matter of common sense that beginning with

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

and reversing the process, will give us

$y=ax^{n}$.

So, when we are dealing with a multiplying constant, we must simply put the constant as a multiplier of the result of the integration.