More Notation for Integrals¶

We will use part 2 of fundamental theorem of calculus a lot, so I introduce some notation to help with it.

New notation for $F(b)-F(a)$¶

Calculating an integral might look like this: $$\begin{array}{rl}{\int }_1^2(x^2+1)dx& =\frac{1}{3}{2}^3+2-(\frac{1}{3}{1}^3+1)\\ & =\frac{10}{3}\end{array}$$ Here I have chosen $F(x)=\frac{1}{3}{x}^3+x$ and calculated $F(2)-F(1)$. It works because $x^2+1$ is continuous on $[1,2]$ and $$\frac{d}{dx}(\frac{1}{3}{x}^3+x)=x^2+1.$$ However, if I forget to mention $F(x)=\frac{1}{3}{x}^3+x$, it can be hard to guess just by looking at the calculation. To fix this problem, we use $[F(x){]}_a^b$ to denote $F(b)-F(a)$. With that, the calculation looks like this: $$\begin{array}{rl}{\int }_1^2(x^2+1)dx& ={[\frac{1}{3}{x}^3+x]}_1^2\\ & =\frac{1}{3}{2}^3+2-(\frac{1}{3}{1}^3+1)\\ & =\frac{10}{3}\end{array}$$ Now it's easy to see which function $1$ and $2$ get plugged into, and that makes it much easier to check that the calculation is correct; just calculate the derivative of what's between $[$ and $]$, and make sure you get $x^2+1$.

There are many other notations for $F(b)-F(a)$. Here are some of the ways to write it I have seen. Pick your favorite notation and use it, or use whatever your teacher or lecturer uses. $$\begin{array}{r}F(b)-F(a)=[F(x){]}_a^b=F(x){|}_a^b={|}_a^{b}F(x)=\underset{a}{\overset{b}{/}}F(x)\end{array}$$

Notation for antiderivatives¶

To denote "a function whose derivative is $f$", we do $$\int f(x)dx.$$ Note that this differs from ${\int }_a^b$ by lacking $a$ and $b$. When you see the integral sign, make sure to check whether it's ${\int }_a^b$ or just $\int$; that completely changes the meaning.

Because of this notation, many people say "integral of $f$" when they mean a function whose derivative is $f$. Usually I write "antiderivative of $f$" instead, just to be clear.

For example, because $\frac{d}{dx}{x}^2=2x$, we can write $$\int 2xdx=x^2.$$ However, because $\frac{d}{dx}(x^2+5)=2x$, we could just as well write $$\int 2xdx=x^2+5.$$ In general, $$\frac{d}{dx}(x^2+\text{constant})=x^2+5$$ for any constant. There are no other antiderivatives than the ones you get by adding constants to $x^2$; we saw it in this derivation. That's why we write $$\int 2xdx=x^2+C,$$ where $C$ denotes an arbitrary constant. In the above equation, $\int f(x)dx$ actually means "all antiderivatives of $f$", and from now on, that's what $\int f(x)dx$ means. Generally, if $F$ is one antiderivative of $f$, then $$\int f(x)dx=F(x)+C.$$

The image is from here.