Write out the form of the partial fraction decomposition of the function: Determine the numerical values of the coefficients, and , where and
Answer
Solving the Integral Using Partial Fraction Decomposition
🧮 Given Expression:
Q = ∫ from A to 11 of (9x / (x² + 4x + 4)) dx
We start by simplifying the denominator:
x² + 4x + 4 = (x + 2)²
🔎 Step 1: Partial Fraction Decomposition
The integrand becomes:
9x / (x + 2)² = A / (x + 2) + B / (x + 2)²
Multiply both sides by (x + 2)² to eliminate denominators:
9x = A(x + 2) + B
Expand the right-hand side:
9x = Ax + 2A + B
📘 Step 2: Compare Coefficients
Match both sides of the equation:
- Coefficient of x: 9 = A
- Constant term: 0 = 2A + B
✅ Solving the System:
A = 9
0 = 2(9) + B → B = -18
0 = 2(9) + B → B = -18
🎯 Final Answer:
A = 9
B = -18
B = -18
Note: This decomposition helps evaluate the integral easily by integrating the simpler rational terms.