The Main Idea 

Think of partial fraction decomposition as taking apart a complicated fraction to reveal the simple pieces hiding inside. Instead of combining fractions like [latex]\frac{1}{x+1} + \frac{2}{x-2}[/latex], you’re doing the reverse—starting with something messy like [latex]\frac{3x}{x^2-x-2}[/latex] and splitting it back into those manageable pieces.

You can only use partial fractions when the degree of the numerator is less than the degree of the denominator. If it’s not, you must use polynomial long division first to create a “proper” fraction.

Problem-Solving Strategy:

1. Check the degree requirement – Is deg(numerator) < deg(denominator)?
2. If not, use polynomial long division to get [latex]\frac{P(x)}{Q(x)} = A(x) + \frac{R(x)}{Q(x)}[/latex]
3. Factor the denominator completely
4. Set up partial fractions based on the factors
5. Solve for the unknown constants
6. Integrate each simple fraction

You can always verify your decomposition by finding a common denominator and checking that you get back to the original fraction.

The Main Idea 

When your denominator breaks down into separate linear factors (like [latex](x-1)(x+3)(2x-5)[/latex]), you get the simplest partial fraction setup. Each distinct linear factor gets its own fraction with a constant numerator—no sharing, no complications.

If [latex]Q(x) = (a_1x + b_1)(a_2x + b_2)\cdots(a_nx + b_n)[/latex] where each factor appears exactly once, then: [latex]\frac{P(x)}{Q(x)} = \frac{A_1}{a_1x + b_1} + \frac{A_2}{a_2x + b_2} + \cdots + \frac{A_n}{a_nx + b_n}[/latex]

Two Methods to Find the Constants:

Method 1 – Strategic Substitution (Usually Faster): Substitute values of [latex]x[/latex] that make individual factors zero. This isolates one constant at a time. For [latex]x(x-2)(x+1)[/latex], try [latex]x = 0[/latex], [latex]x = 2[/latex], and [latex]x = -1[/latex].

Method 2 – Equating Coefficients (More Systematic): Expand everything, collect like terms, and match coefficients on both sides. This gives you a system of equations to solve.

Problem-Solving Strategy:

1. Factor the denominator completely
2. Set up the partial fraction template (one fraction per distinct factor)
3. Find the constants using either method
4. Integrate each simple fraction separately

Always verify your decomposition by adding the fractions back together—you should get your original function.

The Main Idea 

When a linear factor appears multiple times in your denominator, you can’t just use it once in your partial fraction setup. If you have [latex](ax+b)^n[/latex], you need to include all powers from [latex]1[/latex] up to [latex]n[/latex]. Think of it like building a complete ladder—you can’t skip rungs.

For a repeated factor [latex](ax+b)^n[/latex], your decomposition must include: [latex]\frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2} + \frac{A_3}{(ax+b)^3} + \cdots + \frac{A_n}{(ax+b)^n}[/latex]

Don’t forget the lower powers! It is a common mistake to only include the highest power [latex](ax+b)^n[/latex] and miss the essential terms with smaller exponents.

Finding the Constants – Your Two Options:

Strategic Substitution: Choose [latex]x[/latex]-values that zero out specific factors. Then use any other [latex]x[/latex]-value (like [latex]x = 0[/latex]) to find remaining unknowns.

Equating Coefficients: Expand both sides completely, collect like terms by powers of [latex]x[/latex], and match coefficients. This gives you a system of equations to solve systematically.

Remember these key integrals for the final step:

• [latex]\int \frac{1}{ax+b} dx = \frac{1}{a}\ln|ax+b| + C[/latex]
• [latex]\int \frac{1}{(ax+b)^n} dx = \frac{1}{a} \cdot \frac{(ax+b)^{1-n}}{1-n} + C[/latex] for [latex]n \neq 1[/latex]

The Main Idea 

Partial fraction decomposition follows a predictable pattern once you know how to read the denominator. Think of it as having a toolkit where each type of factor in the denominator tells you exactly which “tool” to use in your decomposition.

Problem-Solving Strategy:

1. Check the degree requirement – If deg(numerator) ≥ deg(denominator), use polynomial long division first
2. Factor the denominator completely into linear and irreducible quadratic factors
3. Set up your partial fraction template based on what factors you found
4. Solve for the unknown constants using strategic substitution or equating coefficients
5. Integrate each piece using standard techniques

The Factor-to-Fraction Translation Guide:

Distinct Linear Factor: [latex](ax+b)[/latex] gives [latex]\frac{A}{ax+b}[/latex]

Repeated Linear Factor: [latex](ax+b)^n[/latex] gives [latex]\frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2} + \cdots + \frac{A_n}{(ax+b)^n}[/latex]

Irreducible Quadratic: [latex](ax^2+bx+c)[/latex] gives [latex]\frac{Ax+B}{ax^2+bx+c}[/latex]

Repeated Irreducible Quadratic: [latex](ax^2+bx+c)^n[/latex] gives [latex]\frac{A_1x+B_1}{ax^2+bx+c} + \frac{A_2x+B_2}{(ax^2+bx+c)^2} + \cdots[/latex]

 An irreducible quadratic has no real zeros—check this using the discriminant [latex]b^2-4ac < 0[/latex].