# Partial Fractions

### Partial Fractions Method for Indefinite Integrals:

• Principle: Used for decomposing a complex rational function into simpler fractions.
• Applicability: Typically applied when integrating rational functions (ratio of polynomials).
• Steps:

### Steps for Partial Fractions Decomposition:

1. Proper Fraction:

• Ensure that the degree of the numerator is less than the degree of the denominator.
2. Factorization:

• Factorize the denominator into irreducible factors.
3. #### Partial Fractions Decomposition:

• Express the given rational function as a sum of partial fractions.
• If the factors are linear and distinct: $\frac{P\left(x\right)}{Q\left(x\right)}=\frac{A}{x-{r}_{1}}+\frac{B}{x-{r}_{2}}+\cdots +\frac{N}{x-{r}_{n}}$
• If there are repeated factors: $\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{A}_{1}}{x-{r}_{1}}+\frac{{A}_{2}}{\left(x-{r}_{1}{\right)}^{2}}+\cdots +\frac{{A}_{k}}{\left(x-{r}_{1}{\right)}^{k}}$
4. Determining Coefficients:

• Find the unknown coefficients by various methods like equating coefficients, cover-up method, or simultaneous equations.
5. Integration:

• Integrate each partial fraction term individually.

### Example:

Let's solve the indefinite integral ∫ (4x + 7) / (${x}^{2}$+ 3x + 2) dx using the partial fractions method:

### Steps:

1. Proper Fraction Check:

• The degree of the numerator (1) is less than the degree of the denominator (2). It's a proper fraction.
2. Factorization:

• Factorize the denominator: ${x}^{2}+3x+2=\left(x+1\right)\left(x+2\right)$.
3. Partial Fractions Decomposition:

• Express $\left(4x+7\right)\mathrm{/}\left({x}^{2}+3x+2\right)$ as:

$\frac{4x+7}{{x}^{2}+3x+2}=\frac{A}{x+1}+\frac{B}{x+2}$

4. Determining Coefficients:

• Multiply both sides by the denominator to solve for A and B.
• $4x+7=A\left(x+2\right)+B\left(x+1\right)$
• Substitute values of x to solve for A and B. (e.g., x = -1 and x = -2)

Solving for A and B gives: A = 3 and B = 1.

5. Integration:

• Rewrite the integral in terms of the partial fractions:

$\int \frac{4x+7}{{x}^{2}+3x+2}dx=\int \frac{3}{x+1}+\frac{1}{x+2}dx$

• Integrate each term individually:

$=3\cdot \mathrm{ln}\mathrm{\mid }x+1\mathrm{\mid }+\mathrm{ln}\mathrm{\mid }x+2\mathrm{\mid }+C$

### Tips for Partial Fractions:

• Factorization is Key: Properly factorize the denominator before decomposing into partial fractions.
• Systematic Approach: Write down the partial fraction form and solve for unknown coefficients methodically.
• Check your Work: Ensure the decomposition is correct by recombining the partial fractions.