THE INDEFINITE INTEGRAL
1. THE DERIVED FUNCTION

The derivative of a function at a point can be defined as the following limit

  or lim2.gif (1565 bytes) when h = x-x0

If a function y=F(x) can be differentiated in its domain, it is possible to define a new function, which we will call the derived function, denoted by y=F'(x), and which associates the derivative of the function F to each real number in the domain x0 at the point x0.
1.- Use the rules of differentiation to find the formula of the derived function of the function F(x)=0.25x2-3 

2.- Find the values of F'(-1), F'(0) and F'(2) using the definition of the derivative as a limit. Compare it to the procedures given in activity 1. 

Remember that the derivative of a function when x0 can be interpreted geometrically as the gradient of the tangent to the graph of the function at point (x0,f(x0))

 

The gradient of a line can be found by calculating the difference in height between a point on the line and a point whose x-coordinate is one unit greater.

3.- Find the formula of the derived function of the function  F(x)=0.25x2-3 using the definition of the derived function at a point for any generic point x in the domain.

4.- Change the value of C and note that several functions have the same derived function. What is the relationship between these functions which have the same derivative?
2.  THE REVERSED PROBLEM

Given the function y=f(x), we want to find another function F whose derivative is f. I.e. F'(x)=f(x). Function F is called the antiderivative of function f.
5.- We know the value of f(x) for each value of x as it is the same as the derivative of the function F at x. Can we draw the graph of F by using the tangents?

Use the blue arrows to change the value of x0

6.- Compare the results to those obtained iin the window above. Could we have found a different antiderivative?
3. THE AMBIGUITY OF THE ANTIDERIVATIVE OF A FUNCTION

Now that we have completed the two activities above it seems reasonable to want to answer these two questions:

  1. Is there only one antiderivative of a function?
  2. If there were two different antiderivatives of the same function, what would be the relationship between them?
7.- Deduce that by following the process of reasoning given in exercise 4, if several different functions have the same derivative then reciprocally, one function could have more than one antiderivative.

Point P changes its position by either changing the coordinates of P or by dragging it.

8.- Change the position of P and the parameter x0. Look carefully at the results and work out the importance of point P on the results.

A function f can have infinite antiderivatives and each one is determined by a point which the graph passes through.

4. THE INDEFINITE INTEGRAL

The indefinite integral is defined as the set of all the antiderivatives of the function f. It is represented by the expression indefinida.gif (1042 bytes). It reads as the integral of f with respect to x. The symbol used at the start of the expression (which looks like a stretched s) is called the integral sign and indicates that what comes afterwards is being integrated.
9.- Find the indefinite integral of the function constant to zero (f(x)=0).

Use this window to help you find the solution to activity 9.

10.- Use the activity above to show that two antiderivatives of a function f differ by one constant, such that F(x)=G(x)+k. Clue: First try F-G as an antiderivative of the function 0.

The results above give us a practical way of finding the indefinite integral of a function. We just need to find an antiderivative F(x), and the indefinite integral is the set of all the functions we get from adding any constant to the antiderivative F.

Thus indefinida2.gif (1177 bytes)
11.- Use the linear property of the derivative (the derivative of the sum of functions is the sum of the derivatives of each function and the derivative of a number multiplied by a function is the real number multiplied by the derivative of the function) to show the following properties of the indefinite integral:
  • lineal1.gif (1471 bytes)
  • lineal2.gif (1457 bytes)
  • lineal3.gif (1295 bytes)    where a is a real number.
Enrique Martínez Arcos
Spanish Ministry of Education. Year 2001

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