How do you use partial fraction decomposition to decompose the fraction to integrate #(x^2-5x+6)/(x^3-x^2+2x)#?

Question

Luke Alvarez · Accepted Answer

See below

The denominator factors as: #x(x^2-x+2)# and cannot be factored further using real numbers (discriminant is negative). The integrand cannot be reduced, so we proceed: Find, #A, B, and C# to make: #A/x +(Bx+C)/(x^2-x+2) = (x^2-5x+6)/(x(x^2-x+2))# We need: #Ax^2-Ax+2A+Bx^2+Cx = x^2-5x+6# Solve the system: #A+B=1# #-A+C=-5# #2A=6# Obviously, #A=3# and this makes #B=-2# and #C=-2# The partial fraction decomposition is: #3/x -(2x+2)/(x^2-x+2) = (x^2-5x+6)/(x(x^2-x+2))#. If you have time check by combining the fractions on the left to make sure you've made no mistakes. Now integrate: #int (x^2-5x+6)/(x^3-x^2+2x)) dx = int (3/x)dx - int (2x+2)/(x^2-x+2) dx# The first integral is easy. I'd split the second using: #(2x+2)/(x^2-x+2) = ((2x-1) +3)/(x^2-x+2) # because #int (2x-1)/(x^2-x+2) dx# is a simple substitution and then #int 3/(x^2-x+2) dx# is an inverse tangent with some constants to figure out by completing the square and substituting.

Emily Ammerman · Answer

undefined To decompose the fraction $(x^2 - 5x + 6) / (x^3 - x^2 + 2x)$ using partial fraction decomposition, follow these steps:

1. Factor the denominator $x^3 - x^2 + 2x$ if possible. In this case, the denominator doesn't factor further.

2. Write the fraction as a sum of simpler fractions with unknown numerators:

$\frac{x^2 - 5x + 6}{x^3 - x^2 + 2x} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 1}$

3. Clear the denominators by multiplying both sides of the equation by the original denominator $x^3 - x^2 + 2x$:

$x^2 - 5x + 6 = A(x - 2)(x + 1) + Bx(x + 1) + Cx(x - 2)$

4. Expand the right side and collect like terms:

$x^2 - 5x + 6 = A(x^2 - x - 2) + B(x^2 + x) + C(x^2 - 2x)$

5. Equate the coefficients of like terms on both sides of the equation:

For $x^2$ terms: $1 = A + B + C$
For $x$ terms: $-5 = -A + B - 2C$
For constant terms: $6 = -2A$

6. Solve the system of equations for $A$, $B$, and $C$.

Solving these equations will give you the values of $A$, $B$, and $C$. Once you find those values, substitute them back into the partial fraction decomposition:

$\frac{x^2 - 5x + 6}{x^3 - x^2 + 2x} = \frac{-3}{x} + \frac{2}{x - 2} + \frac{1}{x + 1}$

After decomposing the fraction, you can integrate each term separately.