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

working backwards, we get

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

it is a matter of common sense that beginning with

and reversing the process, will give us

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