Solving Antiderivative of a Function using Table of Integrals

Before anything else, we will start with definitions.

  • Antiderivative

Given a well-defined function $f$ on an interval $[a,b]$, the antiderivative of $f$, which we denote as $F$, is also a function satisfying $F'(x) = f(x)$ on $[a,b]$.

Now, keep in mind that the antiderivative of a function is not unique. We state this fact as a theorem.

Theorem. If $F$ is an antiderivative of $f$ on the interval $[a,b]$, then every antiderivative of $f$ can be written in the form $F(x) + C$, where $C$ is a constant.

Proof. Consider two well-defined functions $f(x)$ and $g(x)$ such that $f'(x) = g'(x)$. Then, \begin{align*}f'(x) &= g'(x) \\ f'(x) - g'(x) &= 0 \\ (f(x) - g(x))' &= (C)' \\ f(x) - g(x) &= C \\ f(x) = g(x) + C\end{align*} Therefore, two functions having the same derivative differ by a constant. $\blacksquare$

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