Page:Calculus Made Easy.pdf/218

 Examples

(1) Given $$\dfrac{dy}{dx} = 24x^{11}$$. Find $$y$$. Ans. $$y=x^32+C$$.

(2) Find $$\int (a + b)(x + 1)\, dx$$. It is $$(a + b) \int (x + 1)\, dx$$ or $$(a + b) \left[\int x\, dx + \int dx\right]$$ or $$(a + b) \left(\dfrac{x^2}{2} + x\right) + C$$.

(3) Given $$\dfrac{du}{dt} = gt^{\frac{1}{2}}$$. Find $$u$$. Ans. $$u = \frac{2}{3} gt^{\frac{3}{2}} + C$$.

(4) $$\dfrac{dy}{dx} = x^3 - x^2 + x$$. Find $$y$$.

and $$\quad \quad \quad \quad y = \tfrac{1}{4} x^4 - \tfrac{1}{3} x^3 + \tfrac{1}{2} x^2 + C$$.

(5) Integrate $$9.75x^{2.25}\, dx$$. Ans. $$y = 3x^{3.25} + C$$.

All these are easy enough. Let us try another case.

Let $$\quad \quad \quad \quad \dfrac{dy}{dx} = ax^{-1}$$.

Proceeding as before, we will write

Well, but what is the integral of $$x^{-1}\, dx$$?

If you look back amongst the results of differentiating $$x^2$$ and $$x^3$$ and $$x^n$$, etc., you will find we never got $$x^{-1}$$ from any one of them as the value of $$\dfrac{dy}{dx}$$. We got $$3x^2$$ from $$x^3$$; we got $$2x$$ from $$x^2$$; we got $$1$$ from $$x^1$$ (that is, from $$x$$ itself); but we did not get $$x^{-1}$$ from $$x^0$$, for two very good reasons. First, $$x^0$$ is simply =1, and is a constant, and could not have