How do you integrate #[((3x^2)+3x+12) / ((x-5)(x^2+9))]dx# using partial fractions?

Question

Aurora Alvarado · Accepted Answer

Please see the explanation

Expand the fraction: #(3x^2 + 3x + 12)/((x - 5)(x^2 + 9)) = A/(x - 5) + (Bx + C)/(x^2 + 9)# Multiply both sides by the left side denominator: #3x^2 + 3x + 12 = A(x^2 + 9) + (Bx + C)(x - 5)# Let x = 5 to make B and C disapper #3(5)^2 + 3(5) + 12 = A(5^2 + 9)# #102 = A(34)# #A = 3# Substitute 3 for A: #3x^2 + 3x + 12 = 3(x^2 + 9) + (Bx + C)(x - 5)# Let x = 0 to make B disappear: #12 = 3(9) + C(-5)# #-15 = -5C# #C = 3# Substitute 3 for C: #3x^2 + 3x + 12 = 3(x^2 + 9) + (Bx + 3)(x - 5)# Let x = 1: #3 + 3 + 12 = 3(1 + 9) + (B + 3)(1 - 5)# #18 = 30 -4B - 12# #B = 0# #int((3x^2 + 3x + 12)/((x - 5)(x^2 + 9)))dx = 3int1/(x - 5)dx + 3int1/(x^2 + 5)dx# #int((3x^2 + 3x + 12)/((x - 5)(x^2 + 9)))dx = 3ln|x - 5| + 3sqrt(5)/5tan^-1((sqrt(5))x /5) + C#

Adam Adkins · Answer

undefined To integrate the given expression using partial fractions, follow these steps:

1. Factor the denominator: (x - 5)(x^2 + 9).
2. Express the fraction as a sum of partial fractions with undetermined coefficients: A/(x - 5) + (Bx + C)/(x^2 + 9).
3. Clear denominators by multiplying both sides by the common denominator.
4. Equate coefficients of like terms on both sides of the equation.
5. Solve for the unknown coefficients A, B, and C.
6. Once you have the partial fractions, integrate each term separately.
7. Finally, sum up the integrals to find the overall integral of the original expression.

The detailed steps would involve solving for A, B, and C by equating coefficients, integrating each term, and then summing them up. If you need further clarification on any of these steps, feel free to ask.