**Indefinite integrals**

[On this course, we will only deal with continuous real functions. Often, continuity is not specifically mentioned in the text, but keep in mind that the results covered during the course are generally, only valid for continuous functions.]

In previous courses, we have learned how to derive functions. In this case, it is said that upon derivation, the derivative function* f'(x)* of the function *f (x)* is obtained. For example:

Sometimes the question is the other way around, e.g. What function has the derivative function *g(x) = 2x*?

The answer, is of course,

It is said that then *f(x)* is an integral function of the function *g(x)*. The different terms are easily confused at first, so lets practice a little more.

If *g(x) = 2x*, its derivative function is *g'(x) = 2*

The derivative function of* f(x)* is* g(x)* and the derivative function of *g(x)* is *h(x)*. The integral function of *h(x) *is *g(x) *and the integral function o*f g(x) *is* f(x)*. Function *g (x)* is at the same time a derivative function of* f(x)* and an integral function of *h(x)*. Just like Leena can be both Maija's daughter and Anna's mother. Anna is Maija's granddaughter in the same way that* h(x) *is the second derivative function of* f(x).*

Finding an integral function is an inverse operation to finding a derivative function. However, one consideration needs to be made:

The derivative gives the same result regardless of the constant term of* f(x)*. That is, all the different functions* f(x)* above are integral functions of the function* f’(x) = 2x*. Indeed, each function has an infinite number of integral functions, one for each possible constant term. This is described by an integration constant, usually denoted by the letter* C*:

Here are all the integral functions of the function *2x.* To get a single integral function, choose the constant *C*. In addition, there is often a notation in which the integral function of the function *f (x)* is denoted by a capital letter* F(x)*. Then* F’(x) = f(x).*

Note that when deriving the function* f (x)*, the derivative function* f’(x)* must be defined unambiguously. When integrating a function, on the other hand, you do not get an unambiguous answer, but you still need to define an integration constant. Usually some additional condition is used for this, as in the following example.

**Example 1**: Find the integral function* F(x)* of the function* f(x) = 4x* when we know that* F(0) = 5.*

First we find a function whose derivative function is *4x*

Finding the integral functions of a function is called integration and is denoted by a stylized* S*-sign (we will explain why later). The previous example can therefore also be written:

At the end of the integral is *dx. *This indicates by which variable the integral is calculated. Most often the variable is *x*, but sometimes it can be something else

**Example 2:**

Integral functions can be found by guessing, as in these examples, and then checking the result by deriving the right side of the equation. It will make things faster and easier if we introduce some integration rules.

These include e.g. removing the constant from under the integral, and the integral rule of the sum:

Moreover, just like with derivation rules, we have our own integration rule for each different function. We will review these in the following chapters one at a time.

**Turn on the subtitles if needed**