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Lecture 15

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MATH 2B - Lecture 15 - Integration of Rational Functions by Partial Fractions

We learn how to integrate any rational function (ratio of polynomials) by expressing it as

a sum of several simpler-functions that we already know how to integrate. These

simpler functions are called partial fractions.

Example:

Solution:

● Solving the left-hand side, we obtain the answer:

○

● Solving the right-hand side directly is not easy:

○ In general, we consider a rational function

■

P(x), Q(x) are polynomials

○ We can only break f(x) into sum of partial fractions when

(deg: degree of polynomials)

○ If , then f(x) is improper, we then need the long division

to divide Q into P until a remainder R(x) is obtained such that

■

(long division)

○ Now, let’s try to solve the integral on the right-hand side by using partial

fractions

1. Factor denominator Q(x)

2. Sum of partial fractions

We assume f(x) can be expressed as:

3. Compute unknown A, B by adding partial fractions

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