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Indefinite Integrals

Indefinite Integrals

At a Glance - Integration by Partial Fractions


Integration by partial fractions is a technique we can use to integrate rational functions when the degree of the numerator is less than the degree of the denominator. Here's the big picture:

  • We start out with an integral whose integrand is a rational function, like

    The degree of the numerator must be less than the degree of the denominator.
     
  • We do some sneaky stuff to rewrite the original rational function as a sum of partial fractions:

     
  • We integrate the partial fractions, whose antiderivatives all involve the natural log:

Be Careful: When x occurs in a denominator with a coefficient other than 1, you have to use integration by substitution.

Example 1

Decompose

into a sum of the form

.


Example 2

Decompose

into partial fractions.



Example 3

Decompose  into partial fractions.


Example 4

Find


Example 5

Find

given that


Exercise 1

Without a calculator, find


Exercise 2

Find


Exercise 3

Find the sum.


Exercise 4

Decompose into partial fractions.


Exercise 5

Decompose into partial fractions.


Exercise 6

Decompose into partial fractions.


Exercise 7

Decompose into partial fractions.


Exercise 8

Decompose into partial fractions.


Exercise 9

Decompose into partial fractions.


Exercise 10

Decompose into partial fractions.


Exercise 11

Decompose into partial fractions.


Exercise 12

Decompose into partial fractions.


Exercise 13

Decompose into partial fractions.


Exercise 14

Integrate.


Exercise 15

Integrate.


Exercise 16

Integrate.


Exercise 17

Integrate.


Exercise 18

Integrate.


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