Primitive function/Definite integral/Examples and remarks/Section

From Wikiversity
Jump to navigation Jump to search

Let denote a real interval, and

a primitive function for a function . Suppose that . Then one sets

This notation is basically used for computations, in particular, when we want to determine definite integrals.

Using known results about the derivatives of differentiable functions, we obtain a list of primitive functions for some important functions. In general however, it is difficult to find a primitive function.

The primitive function of , where and , , is .

Suppose that the distance between two masses (thought of as mass points) and is . Because of gravitation, this system contains a certain potential energy. How is this potential energy changing, when we move these masses to a distance ?

The needed energy is force times path, where the force itself depends on the distance between the masses. Due to the gravitation law, the force, given the distance between the masses, equals

where denotes the constant of gravitation. Therefore, the energy needed to increase the distance from to , equals

Hence it is possible to assign a value to the difference between the potential energies for the two distances and , though it is not possible to assign an absolute value to the potential energy for a given distance.

The primitive function of the function is the natural logarithm.

The primitive function of the exponential function is the exponential function itself.

The primitive function of is , the primitive function of is .

The primitive function of is , due to fact.

The primitive function of (for ) is , because we have

Caution! Integration rules are only applicable for functions, which are defined on the whole interval. In particular, the following is not true

since we integrate over a point where the function is not defined.

We consider the function

given by

This function is not Riemann-integrable, because it it neither bounded from above nor from below. Hence, there exist no upper step functions for . However, still has a primitive function. To see this, we consider the function

This function is differentiable. For , the derivative is

For , the difference quotient is

For , the limit exists and equals , so that is differentiable everywhere (but not continuously differentiable). The first summand in is continuous, and therefore, due to fact, it has a primitive function . Hence is a primitive function for . This follows for from the explicit derivative and for from