Integration of standard functions

Summary and examples

In the following formula is called the primitive of the integrand :

where satisfies:

 a. (constante) b. c. d. e. f. g.

Furthermore the following rules hold:

h.

Thus far the integrals are so-called indefinite integrals, meaning that the integral sign has no boundaries. This is in contrast to definite integrals which have an upper and a lower boundary.

For a definite integral we have:

i.

In the following examples we show how we calculate integrals. The specific rule is also given.

Example 1

Solve:

The primitive function is calculated by applying a. and thus:

Example 2

Solve:

The primitive is calculated by applying a. in combination with h. an thus:

Example 3

Solve:

The primitive is calculated by applying b. and thus:

Example 4

Solve:

The primitive is calculated by applying e. in combination met h. and thus:

Example 5

Solve:

The primitive is calculated by applying f. in combination with h. and i. and thus:

Remember, .

Example 6

Solve:

The primitive is calculated by applying a. and g. in combination with h. and i. and thus:

Example 7

Solve:

The primitive is calculated by applying c. in combination with i. and thus:

Example 8

Solve:

This function does not seem to appear in the list of standard functions, but indirectly this is still the case. We can write:

and thus rule a. can be applied with .

So:

Example 9

Solve:

This function does not seem to appear in the list of standard functions, but indirectly this is still the case. We can write:

and thus (by applying a., h. and i.):

=

Example 10

Solve:

This would seem an awkward integral, but the integrand (the function under the integral sign) can be written as follows:

and thus the integral becomes:

Here a., b. and h. are applied.

Example 11

Solve:

We can write the integrand (the function under the integral sign):

and thus the integral becomes (with a. and h.):

0