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Integral calculus

Here we summarize everything important on the topic Integral calculus together and explain it to you Note boxes, Step-by-step instructions and Examples!

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Integral calculus explained simply

Integral calculus is a branch of Analysisthat is closely related to the Differential calculus is linked. Just as differential calculus is primarily about determining the derivative of a function, integral calculus deals with determining a Indefinite integral and the statements that can be drawn from it.

An integral has the following form, the designations are assumed to be known in the following.

In this text we want to introduce all the important chapters on integral calculus one after the other and explain the most important information and calculation rules to you in a clear and logical way.

Indefinite integral

The determination of an antiderivative is the central topic of integral calculus and serves as the basis for all subsequent chapters. An antiderivative is defined as follows:

The function called Indefinite integral of , if

It is very practical that every continuous function such an antiderivative owns. But how can you calculate it?

For this you use the HDI, that's the one Law of differential and integral calculus. It establishes the connection between deriving and integrating.

With the above definition it would be easy if you determine the antiderivative have given. You just have to deduce. Now we want to reverse this process, so to speak, you can understand integrating (deriving) as the reverse of deriving!

That means, you can calculate an antiderivative directly via the integral:

example

Find an antiderivative of . So we are looking for a function that just derived results. For this we calculate

Now we have to consider what is derived would result and see immediately (taking into account the derivation rules) that

Note: The constant stands for any real number that is used when deriving falls away. So you can see right away that it is infinitely many antiderivatives to there, depending on what you are for begins.

Like the primitives for all important functions look and how you calculate them, we explain in detail in a separate article on the antiderivatives.

The above definition of the antiderivative is very application-oriented and shows you how best to calculate it. It can be defined somewhat more theoretically using the integral function.

Integral function

An integral function is a function that has the following form:

The difference to the general antiderivative is that here a certain integral is considered, with a lower limit and the variables as an upper limit. So here you are calculating a concrete antiderivative that is in the point has a zero.

Note: Any integral function has at its lower integration limit a zero, i.e.

Note: Every integral function is an antiderivative, but not every antiderivative is also an integral function. This is especially true because an antiderivative can be shifted up / down at will.

Definite integral and indefinite integral

As you have just seen with the difference between an integral function and an antiderivative, there are two types of integrals in integral calculus, namely the definite and the indefinite integral.

The indefinite integrals