# Integration of standard functions

## Summary and examples

In the following is called the primitive of : where satisfies: f(x) F(x)  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