Integration by Partial Fractions

in mathematics •  last month  (edited)

In this video I go over an introduction on the method of partial fractions for integrating rational functions. Basically rational functions can be simplified by breaking them down to partial fractions using partial fraction decomposition which in turn greatly simplify the integration process. For an in depth look into partial fraction decomposition make sure to watch my earlier videos shown in the links below.

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Integration of Rational Functions by Partial Fractions

Integration by Partial Fractions.jpeg

One method of integrating rational functions (which are simply ratios of polynomials) is by expressing them as a sum of simpler functions, called partial fractions, that we already know how to integrate. To illustrate the method consider the following:

Now we reverse the procedure we can see how to integrate the function of the right side of the equation:

To see how the method of partial fractions works in general, let's consider a rational function:

where P and Q are polynomials.

It's possible to express f as a sum of simpler fractions provided that the degree of P is less than the degree of Q. Such a rational function is called proper. Recall that if:

where an ≠ 0, then the degree of P is n and we write deg(P) = n.

If f is improper, that is, deg(P) ≥ deg(Q), then we must take the preliminary step of dividing Q into P (by long division) until a remainder R(x) is obtained such that deg(R) < deg(Q). The division statement is:

where S and R are also polynomials.

The partial fractions will be solved using the methods in my earlier videos on partial fraction decomposition.

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