Decomposition of Fractions

Integration in calculus is how we find the area between a curve and the x axis. Examples: vibration, distortion under weight, or one of many types of "fluid flow" -- be it heat flow, air flow (over a wing), or water flow (over a ship's hull, through a pipe, or perhaps even groundwater flow regarding a contaminant) All these things can be either directly solved by integration (for simple systems), or some type of numerical integration (for complex systems). Decomposition of fractions makes the process of integration easier.

Goal: To split this fraction into two or more parts Step 1: Determine if the numerator is linear or quadratic or if the degree of the numerator is greater than the denominator. Step 2: If an improper fraction (the degree of the numerator is greater than the degree of the denominator) must divide numerator by the denominator FIRST. Step 3: Once division is done, if needed, FACTOR the denominator. Step 4: Linear Factors: For EACH factor of the form (px + q) m, the partial fraction decomposition must include the following sum of m fractions. Step 5: Quadratic Factors: For EACH factor of the form (ax 2 + bx + c) n, the partial fraction decomposition must include the following sum of n fractions.

Step 1: Numerator is linear. Step 2: The expression is proper so no need to divide. Step 3: Factor denominator. Step 4: Clear the denominator by multiplying both sides by (x-3)(x+2)

Step 1: x can be any number we want. So let’s pick an x so that it’s easy to solve. Like -2 and 3. Step 2: To solve for A let x = 3 (3) + 7 =A (3 + 2) + B(3 -3) 10 = 5A A = 2 Step 3: To solve for B let x = -2 (-2) + 7 =A (-2 + 2) + B(-2 -3) 5= -5 B B = -1

Step 1: IMPROPER, so must divide! Step 2: Factor denominator. x(x 2 + 2x + 1)Or x(x + 1) 2 Step 3: That squared term means we have to include one partial fraction for each power up to 2.

Step 4: Solve for A,B,C by clearing the denominator and letting x = 0, -1. x Step 5: Solve for C by letting x = -1.

Step 6: Solve for A by letting x = 0. Step 7: We’ve used all the convent x’s so let x equal anything. So letting A = 6 and C = 9, let’s pick x = 1