Can the definite integral be expressed as follows: $\int_{a}^{b}f(x)dx=[F(x)]_{a}^{b}=[\int f(x)dx -C]_{a}^{b} ?$ Where $\int f(x)dx=F(x)+C$

Question

This way of writing the definite integral is useful for some type of questions requiring implicit substitution so I am wondering if it is a valid notation.

Answer 1

Don't write things like this!

If $F_0$ is a primitive of $f$ in an interval containing $[a,b]$ it is alright to write $$\int_a^b f(x)\>dx=F_0(x)\biggr|_a^b\ .\tag{1}$$ Here the LHS is a number, namely the limit of certain Riemann sums computed with $f$, and by the Fundamental Theorem of Calculus it is also $=F(b)-F(a)$ for any primitive of $f$ on $[a,b]$, in particular of $F_0$. That's what the RHS of $(1)$ expresses in condensed form.

On the other hand, for any $f$ we have the general notation $f'$ for the derivative of $f$. In a similar way there is a notation for the "antiderivative" of $f$. Now this "antiderivative" is not a single function, but an infinite set of functions $F:\>[a,b]\to{\mathbb R}$, whereby any two functions of this set differ only by an additive constant. These premises lead to the notation $$\int f(x)\>dx$$ for the full set of antiderivatives. If $F_0$ is a single known antiderivative then it is very common to write $F_0(x)+C$, or similar, for this same set. But this notation is somewhat undue, since $C$ is a dummy variable. The actual meaning is $$\int f(x)\>dx=\bigl\{F\bigm|\exists C\in{\mathbb R}: \ F(x)=F_0(x)+C \ (a\leq x\leq b)\bigr\}\ .$$ At any rate, writing $\ \int f(x)\>dx -C\ $ doesn't make sense.