Important Steps to Resolve Partial Fractions
- Ensure proper fraction:
If degree(P) ≥ degree(Q), perform long division first. - Factorize denominator completely.
- Assign constants (A, B, C, …) according to factor types.
- Multiply both sides by the denominator to eliminate fractions.
- Solve for constants by:
- Substitution (plugging suitable x values), or
- Comparing coefficients.
Partial Fractions — Lecture Notes
SaitechAI Mathematics Lecture Series (Class 11–12, JEE/NEET Level)
1. Definition
A partial fraction expresses a rational function as a sum of simpler fractions. If \( \frac{P(x)}{Q(x)} \) is a rational function and \( \deg P(x) < \deg Q(x) \), it can be written as a sum of partial fractions.
2. Basic Rule
If \( \frac{P(x)}{Q(x)} \) is proper, its decomposition depends on the factors of \( Q(x) \):
- Distinct Linear Factors: \( \frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b} \)
- Repeated Linear Factors: \( \frac{P(x)}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2} \)
- Irreducible Quadratic Factor: \( \frac{P(x)}{(x^2+bx+c)} = \frac{Ax + B}{x^2 + bx + c} \)
3. Steps to Resolve Partial Fractions
- Ensure the fraction is proper; if not, divide first.
- Factorize the denominator completely.
- Assign constants \(A, B, C, \dots\) based on factor types.
- Multiply by the denominator and remove fractions.
- Solve for constants by substitution or comparing coefficients.
4. Examples
Example 1:
\( \frac{3x – 5}{(x – 1)(x + 2)} = \frac{A}{x – 1} + \frac{B}{x + 2} \)
Multiply both sides by \( (x – 1)(x + 2) \):
\( 3x – 5 = A(x + 2) + B(x – 1) \)
Let \( x = 1 \Rightarrow A = -\frac{2}{3} \);
\( x = -2 \Rightarrow B = \frac{11}{3} \)
Final form:
\( \frac{3x – 5}{(x – 1)(x + 2)} = \frac{-2/3}{x – 1} + \frac{11/3}{x + 2} \)
Example 2 (Repeated Factor):
\( \frac{2x + 3}{(x + 1)^2} = \frac{A}{x + 1} + \frac{B}{(x + 1)^2} \)
Multiply: \( 2x + 3 = A(x + 1) + B \)
Let \( x = -1 \Rightarrow B = 1 \); comparing coefficients → \( A = 2 \)
So, \( \frac{2x + 3}{(x + 1)^2} = \frac{2}{x + 1} + \frac{1}{(x + 1)^2} \)
Example 3 (Irreducible Quadratic):
\( \frac{2x^2 + 3x + 4}{(x + 1)(x^2 + 1)} = \frac{A}{x + 1} + \frac{Bx + C}{x^2 + 1} \)
5. Applications
- Integration of rational functions
- Laplace transforms
- Electrical circuit analysis (RC, RL)
- Control systems and differential equations
6. Common Mistakes
- Not dividing when numerator degree ≥ denominator degree
- Missing terms for repeated or quadratic factors
- Incorrect coefficient comparison
7. Quick Practice Problems
- \( \frac{x + 2}{x^2 – 1} \)
- \( \frac{2x + 3}{(x – 1)^2} \)
- \( \frac{3x^2 + 5x + 2}{x(x + 1)(x + 2)} \)
- \( \frac{2x + 1}{x^2 + 4x + 5} \)
8. Integration via Partial Fractions
After decomposition, integrate each term separately:
\( \int \frac{A}{x – a} dx = A \ln|x – a| + C \)
\( \int \frac{Bx + C}{x^2 + px + q} dx = \text{Use substitution or arctan form} \)
