Indefinite integral

We know some concepts related to the derivation: if $$F(x)$$ is a function, we denote $$F'(x)$$ 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, $$F'(x)=f(x)$$.

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 $$\dfrac{d}{dx}C=0$$. Therefore, given $$f(x)$$, with a primitive function $$\dfrac{d}{dx}F(x)=F'(x)=f(x)$$, but them also $$F (x) +C$$ is a valid primitive since we also know that $$\dfrac{d}{dx}(F(x)+C)=f(x)$$. 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:

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