Lec 6 - Integration by Rational Function by Partial Fraction

Sec 7.4

Techniques of Integration

1. Substitution
1. Integration by parts
2. Trig Integrals
1. Trig substitutions
1. Partial functions

Recall,

We will learn how to do this backwards so that the integrand can be rewritten as a sum of terms each of which we can be rewritten as a sum of terms each of which we can integrate.

Definition

We only apply partial fractions on proper rational functions.

If not proper then we long division to reduce it

Example

Evaluate

We can use long division.

Step 1: Use long division to get a proper fraction

Step2: Decompose our proper rational function

In this case we always assume,

The method I present is different from the textbook.

On the LHS, we have

Therefore,

Eg

Evaluate

Step 1: It is proper, no long division.

Step 2: We use partial fractions

Step 3: Evaluate

What if weinstead have

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