Integration of Algebraic Functions
Given a constant and two functions and , two basic properties of integrals are These two properties follow from the differential formulas:
For a real number , the indefinite integral of is where is the constant of integration.
This can easily be shown through an application of the fundamental theorem of calculus:
We know by the power rule that where is an arbitrary constant. Multiplying both sides by gives
Note that as this would make the left-hand side indeterminate. The first fundamental theorem of calculus tells us that differentiation is the opposite of integration. Using this fact, let us take the integral of both sides: As stated above, is an arbitrary constant, so we can set as anything as long as . Letting where we have
Evaluate the integral .
We have Applying our theorem from above, where is the constant of integration.
As stated above, we can let be any arbitrary real number as long as . This means that our formula works not only for whole numbers but for negative numbers, rational numbers, and irrational numbers as well. For more information on the integral of , please refer to the wiki Integration of Rational Functions.
Evaluate the integral
We have Applying our theorem from above, where is the constant of integration.
Evaluate the integral .
We have Applying our theorem from above, where is the constant of integration.
Our theorem can be extended to all polynomial expressions through the application of the properties of integrals.
By the properties of integrals, given a polynomial we have
where is the constant of integration.
Evaluate the integral .
Using the above properties, we have where is the constant of integration.
Evaluate the integral
We have
where is the constant of integration.
If , what is
Note that is a polynomial but is not in the form given in the summary above. We will later see methods to integrate this function directly, but to use the basic properties above, we first expand the polynomial by the binomial theorem. This gives
which implies
where is the constant of integration.