Integration Using Partial Fractions With Linear Factors

Introduction to Partial Fractions

Integration by partial fractions is a technique used to integrate rational functions—which are functions expressed as a quotient of two polynomials, $\frac{N(x)}{D(x)}$. The core idea is to decompose this potentially complex rational function into a sum of simpler fractions that are individually easier to integrate.

This calculator focuses on a common case: where the denominator $D(x)$ can be factored into distinct linear terms (e.g., $(x-a)(x-b)$ where $a \neq b$).

The power of this method lies in transforming a difficult integral into several basic ones. Integrals of simpler fractions like $\frac{A}{x-r}$ or $\frac{A}{px+q}$ are standard forms, typically resulting in natural logarithms: for instance, $\int \frac{A}{px+q} dx = \frac{A}{p} \ln|px+q| + C$.

Method: Distinct Linear Factors

To integrate a rational function $\frac{N(x)}{D(x)}$ using partial fractions, where $D(x)$ has distinct linear factors:

1. Ensure Proper Fraction: The degree of the numerator $N(x)$ must be less than the degree of the denominator $D(x)$. If not (i.e., the fraction is improper), perform polynomial long division first. This will result in a polynomial (or constant) plus a proper rational function, which can then be decomposed. This solver assumes the fraction is proper or has been made proper.
2. Factorize Denominator: Completely factorize the denominator $D(x)$ into its distinct linear factors.

   $$ D(x) = (p_1x + q_1)(p_2x + q_2) \cdots (p_kx + q_k) $$

   Each $(p_ix + q_i)$ is a unique linear factor.
3. Set Up Partial Fraction Form: For each distinct linear factor $(p_ix + q_i)$ in the denominator, there will be a corresponding partial fraction term of the form $\frac{A_i}{p_ix + q_i}$. The original rational function is then expressed as the sum of these terms:

   $$ \frac{N(x)}{D(x)} = \frac{A_1}{p_1x + q_1} + \frac{A_2}{p_2x + q_2} + \cdots + \frac{A_k}{p_kx + q_k} $$

   The constants $A_1, A_2, \ldots, A_k$ are what we need to find.
4. Find the Coefficients $A_i$ (Heaviside Cover-Up Method): This is an efficient way to find the coefficients for distinct linear factors.
   • To find a specific coefficient, say $A_i$, corresponding to the factor $(p_ix + q_i)$:
     1. Determine the root of this factor: $p_ix + q_i = 0 \implies x = r_i = -\frac{q_i}{p_i}$.
     2. In the original fraction $\frac{N(x)}{D(x)}$, "cover up" (mentally or actually remove) the factor $(p_ix + q_i)$ from the denominator $D(x)$. Let the remaining part of the denominator be $D_i(x) = \frac{D(x)}{p_ix+q_i}$.
     3. Substitute the root $x = r_i$ into this new fraction $\frac{N(x)}{D_i(x)}$ to get $A_i$:

        $$ A_i = \frac{N(r_i)}{D_i(r_i)} = \left. \frac{N(x)}{\frac{D(x)}{p_ix+q_i}} \right|_{x=r_i} $$

        This works because if we were to clear denominators in the decomposition by multiplying by $D(x)$, and then substitute $x=r_i$, all terms on the right side except the one containing $A_i$ would become zero.
   • Repeat this process for each coefficient $A_1, A_2, \ldots, A_k$.
5. Integrate the Partial Fractions: Once all coefficients are found, substitute them back into the decomposed form. The integral of the sum is the sum of the integrals:

   $$ \int \frac{N(x)}{D(x)} dx = \int \frac{A_1}{p_1x + q_1} dx + \int \frac{A_2}{p_2x + q_2} dx + \cdots + \int \frac{A_k}{p_kx + q_k} dx $$

   Each of these integrals is straightforward:

   $$ \int \frac{A_i}{p_ix + q_i} dx = A_i \cdot \frac{1}{p_i} \ln|p_ix + q_i| $$

   This comes from a simple u-substitution (let $u = p_ix + q_i$, then $du = p_i dx$, so $dx = \frac{1}{p_i} du$).
6. Combine and Add Constant: Sum the results of the individual integrations and add the constant of integration, $C$.

   $$ \text{Final Result} = \sum_{i=1}^{k} \frac{A_i}{p_i} \ln|p_ix + q_i| + C $$

Limitations & Notes

• This solver assumes the degree of the numerator $N(x)$ is strictly less than the degree of the denominator $D(x)$ (i.e., the rational function is proper). If it's improper, polynomial long division must be performed first to get a polynomial plus a proper rational function. This solver will issue a warning if it heuristically detects an improper fraction but will attempt to proceed.
• The denominator $D(x)$ must be provided as a product of distinct linear factors. Examples: $(x-1)(x+2)$, $(2x+1)(x-5)(x)$.
• Cases not currently handled by this specific solver:
  • Repeated linear factors (e.g., $(x-1)^2$). These require a different setup for partial fractions (e.g., $\frac{A}{x-1} + \frac{B}{(x-1)^2}$).
  • Irreducible quadratic factors (e.g., $x^2+1$). These lead to terms like $\frac{Ax+B}{x^2+1}$ in the decomposition.
• The Heaviside cover-up method is very efficient for distinct linear factors. For more complex cases like repeated factors or irreducible quadratics, a method involving equating coefficients or substituting strategic values of $x$ is generally used.
• Use MathCrave AI math solver to counter this limitation