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.):