Indefinite integral

We know some concepts related to the derivation: if $F(x)$ is a function, we denote $F^{\prime }(x)$ to its derivative and claculate according to the rules already seen.The problem that we want to tackle now is to do the reverse, that is, from a derivative, let's call it $f(x)$, we want to find what function $F(x)$ has as a derivative $f(x)$. Or, $F^{\prime }(x)=f(x)$.

In other words, we write ${\displaystyle \int f(x)dx=F(x)}$, which means that $f(x)$ is the derivative of $F(x)$ with respect to the variable $x$. Then, $F(x)$ is the indefinite integral, primitive function, or antiderivative of $f(x)$.

Let's observe that we use the symbol ${\displaystyle \int }$ to denote that we are integrating, and $dx$ to signify what variable we are integrating. In some cases this $dx$ might be omitted, but to avoid confusion it is better to always use it.

Let's see now some important properties of the indefinite integral:

• We know that the derivative of a constant $C$ is ${\displaystyle \frac{d}{dx}}C=0$. Therefore, given $f(x)$, with a primitive function ${\displaystyle \frac{d}{dx}}F(x)=F^{\prime }(x)=f(x)$, but them also $F(x)+C$ is a valid primitive since we also know that ${\displaystyle \frac{d}{dx}}(F(x)+C)=f(x)$. Therefore, the primitive or antiderivative of a function is not unique. Thus, when computing an integral we will give the result as: ${\displaystyle \int f(x)dx=F(x)+C}$, where $C$ is called an integration constant. Note that we should never forget this constant.

• The integral, as well as the derivative, satisfies the properties of linearity, that is:

  • ${\displaystyle \int k\cdot f(x)dx=k\cdot \int f(x)dx}$
  • ${\displaystyle \int (f(x)+g(x))dx=\int f(x)dx+\int g(x)dx}$