Why is it legal to take the antiderivative of both sides of an equation?

Question

first, I must apologize for somewhat misleading a title.

To save both your and my time, I will go straight to the point.

By definition, an indefinite integral, or a primitive, or an antiderivative of a (some condition) function $f(x)$ is any $F(x)$ such that $F'(x)=f(x)$. All well and good.

Because any other primitive can be written as $F(x)+C$ for some constant $C$ (and this requires a proof), if we were to denote by $\int f(x)dx$ an antiderivative of $f(x)$, then \begin{equation} \int f(x)dx = F(x)+C. \end{equation}

Fine. But here is the part that every textbook seems to have no problem with, but bugs me greatly: Often they say that integrate both sides of the following equation: \begin{equation} f(x)=g(x),\end{equation}

to obtain \begin{equation} \int f(x)dx = \int g(x)dx. \end{equation}

This looks like an ABSOLUTE nonsense to be for the following reason: IF both sides of the previous equation are TRULLY equal, then surely

\begin{equation} \int f(x)dx - \int g(x)dx =0. \end{equation}

But

\begin{equation} \int f(x)dx - \int g(x)dx =\int (f(x)-g(x))dx = \int 0dx, \end{equation}

which then equals $C$, any constant. Surely this is not necessarily 0!

So in short, this is my question: IS IT, STRICTLY SPEAKING, LEGAL, TO TAKE THE ANTIDERIVATIVE OF BOTH SIDES OF AN EQUATION?

Answer 1

The problem here might be the notation $\int f(x)\,dx$. Does it mean one primitive? All primitives? Something else?

If we for a while agree that $F$ is a primitive of $f$ on an interval $I$ and $G$ is a primitive of $g$ on the same interval $I$, and it holds that $f(x)=g(x)$ for all $x\in I$, then we can be sure that $F(x)=G(x)+C$ for all $x\in I$, where $C$ is some constant.

Answer 2

Paul's Online Math Notes has an excellent section on the Constant of Integration. It shows how two apparently different results of integrating the same equation differ only by the Constant. Since we often write the Constant simply as $C$, it's easy to miss the meaning of this term.

While this discussion doesn't deal precisely with the question at hand, it is germane in as much as it deals with nomenclature of apparently different functions while integrating, and the resulting Constant.

Read the whole discussion here.