Integration Using Partial Fractions with Repeated Linear Factors

Integration by partial fractions is a technique used to integrate rational functions, which are functions expressed as a quotient of two polynomials, say $\frac{N(x)}{D(x)}$. The core idea is to decompose this potentially complex rational function into a sum of simpler fractions that are individually easier to integrate. This method is particularly useful when the denominator $D(x)$ can be factored. This calculator focuses on cases where $D(x)$ can be factored into linear terms, which may be distinct (e.g., $(x-1)(x+3)$) or repeated (e.g., $(x-2)^3$). The power of this technique lies in the fact that integrals of these simpler fractions, such as $\frac{A}{(px+q)}$ or $\frac{A}{(px+q)^k}$, have standard integration forms, typically involving natural logarithms or simple power rule applications.

Prerequisite: Degree of Numerator

A crucial first step is to ensure that the degree of the numerator polynomial $N(x)$ is strictly less than the degree of the denominator polynomial $D(x)$. If it's not (i.e., degree of $N(x) \ge$ degree of $D(x)$), polynomial long division must be performed first. This division will result in a polynomial plus a new rational fraction where the numerator's degree is less than the denominator's. The partial fraction technique is then applied to this new (proper) rational fraction.

Let's consider a proper rational function $\frac{N(x)}{D(x)}$. The process involves several key steps:

1. Factorize the Denominator $D(x)$

The first step is to completely factorize the denominator $D(x)$ into its linear factors (and irreducible quadratic factors, though this calculator focuses on linear ones). For example, $D(x) = (x-1)^2(x+2)$.

2. Set Up the Partial Fraction Decomposition Form

Based on the factors of $D(x)$, we write down the form of the partial fraction decomposition with unknown coefficients (A, B, C, etc.).

• For each distinct linear factor of the form $(px+q)$ in $D(x)$, the decomposition includes a term: $$ \frac{A}{px+q} $$
• For each repeated linear factor of the form $(px+q)^m$ in $D(x)$, the decomposition includes $m$ terms: $$ \frac{A_1}{px+q} + \frac{A_2}{(px+q)^2} + \cdots + \frac{A_m}{(px+q)^m} $$

For example, if $\frac{N(x)}{(x-1)^2(x+2)}$, the form would be $\frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+2}$.

3. Solve for the Unknown Coefficients

This is often the most algebraic part. There are several methods:

• Equating Coefficients: Multiply both sides of the decomposition by $D(x)$ to clear denominators. Then, expand and equate the coefficients of corresponding powers of $x$ on both sides. This results in a system of linear equations for the unknown coefficients.
• Substituting Convenient Values of $x$: After clearing denominators, substitute strategic values of $x$ (especially the roots of the factors in $D(x)$) to simplify the equation and solve for some coefficients directly.
• Heaviside Cover-Up Method (for distinct linear factors): For a term $\frac{A}{px+q}$ from a distinct linear factor, $A$ can be found by "covering up" $(px+q)$ in the original fraction $\frac{N(x)}{D(x)}$ and evaluating the rest at $x = -q/p$. So, $A = \left[ \frac{N(x)}{D(x)/(px+q)} \right]_{x=-q/p}$.
• For Repeated Linear Factors: If $D(x)$ has a factor $(px+q)^m$, and we let $r = -q/p$ be the root. Let $G(x) = \frac{N(x)}{D(x)/(px+q)^m}$ (i.e., the original fraction with the $(px+q)^m$ factor removed from the denominator). The coefficients for the terms $\frac{A_k}{(px+q)^k}$ are found using $G(x)$ and its derivatives evaluated at $x=r$:
  • Coefficient for $\frac{1}{(px+q)^m}$ is $A_m = G(r)$
  • Coefficient for $\frac{1}{(px+q)^{m-1}}$ is $A_{m-1} = G'(r)$
  • Coefficient for $\frac{1}{(px+q)^{m-2}}$ is $A_{m-2} = \frac{1}{2!} G''(r)$
  • In general, for $\frac{1}{(px+q)^{m-j}}$, the coefficient is $A_{m-j} = \frac{1}{j!} G^{(j)}(r)$ (where $G^{(j)}(r)$ is the $j$-th derivative of $G(x)$ evaluated at $x=r$).

This calculator primarily uses the method involving derivatives for repeated linear factors and the cover-up method (a special case of the derivative method where $j=0$) for distinct factors.

4. Integrate the Sum of Simpler Fractions

Once all coefficients are found, substitute them back into the decomposition. The original integral now becomes a sum of simpler integrals, which are usually standard forms:

• $\int \frac{A}{px+q} dx = \frac{A}{p} \ln|px+q| + C_1$
• For $k > 1$, $\int \frac{A}{(px+q)^k} dx = A \int (px+q)^{-k} dx = \frac{A}{p} \frac{(px+q)^{-k+1}}{-k+1} + C_2 = \frac{A}{p(1-k)(px+q)^{k-1}} + C_2$

The final result is the sum of these individual integrals, plus a single constant of integration $C$.

This calculator currently supports:

• Rational functions $\frac{N(x)}{D(x)}$ where the degree of the numerator $N(x)$ is strictly less than the degree of the denominator $D(x)$. If this condition is not met, you must perform polynomial long division first and then apply partial fractions to the resulting proper rational function remainder.
• Denominators $D(x)$ that can be factored into linear factors. These linear factors can be distinct (e.g., $(x-1)(x+2)$) or repeated (e.g., $(x-3)^2(2x+1)^3$).
• Input for the denominator should be in a factored form, using * for multiplication between factors, e.g., (x-1)^2 * (x+2).

Not yet supported:

• Irreducible quadratic factors in the denominator (e.g., factors like $x^2+1$ or $x^2+x+5$ that cannot be factored further into real linear factors). Decomposition for these involves terms like $\frac{Ax+B}{ax^2+bx+c}$.
• Automatic polynomial long division.

Solution:

• Use MathCrave AI solver to overcome this limitation