How to find initial function of a function

Question

I am stuck in this problem. How do I find the initial function of a given function? I am learning integral and there in the formula $S=F(b)-F(a)$ is $F$ the initial function of $f$. For example, $f(x)=x^2$, one of its initial functions can be $F(x)=\dfrac{x^3}{3}$. This is what my textbook says but I don't know how this comes and how I can find the initial functions of other functions. :(.

Is there a formula for this?

Answer 1

The rule that your textbook is using is that an antiderivative (what you call the initial function) of $x^n$ is $\frac{1}{n + 1}x^{n + 1}$.

Answer 2

Besides to @Jim's answer, you may find an anti-derivatives of the following elementary functions: $$\text{e}^{ax},~~~\cos(bx),~~~\sin(bx),~~~ x^k$$ by using the derivation well-known formulas and some elementary per-calculus formulas. And may be you cannot find the initial functions via this approach. For example, you can find the anti-derivative functions of $\cos(2x)$ and $\exp(-5x)$ which are $\frac{1}2\sin(2x)$ and $\frac{1}{-5}\text{e}^{-5x}$ respectively; but without using some certain methods you may not able to find the anti-derivative of $\cos(2x)\exp(-5x)$ just by applying the differentiation formulas.