Partial Fractions:

Breaking a rational expression into partial fractions is:

Rules for finding partial fractions:

1. The numerator must be a lower degree than the denominator, if not then divide until the remainder term is in the proper form.
2. The denominator must be factored, so that every factor is either a linear factor or a quadratic factor with real coefficients.
3. This fraction can be broken down into partial fractions, that is dependent upon the factors of the denominator

ex.

Evaluate:

Break into partial fractions:

Multiply (x + 1)(x + 3) to both sides of the equation.

1 = A (x + 3) + B (x + 1)
= Ax + 3A + Bx + B
= Ax + Bx + 3A + B
1 = (A + B)x + 3A + B

The coefficients on both sides of the equation must be the same, that is that the coefficient of x on the left side of the equation must equal the coefficient of x on the right side of the equation.

A + B = 0
3A + B = 1

solve for A and B,

The integral is equal to:

ex.

Break into partial fractions:

3x - 1 = A [ x(x + 1)] + B(x + 1) + Cx²
3x - 1 = (A + C)x² + (A + B)x + B

A + C = 0
A + B = 3
B = 1
A = 2
C = -2

 

= 2 ln |x| - 1/x - 2 ln |x + 1| + C