Integration Using Partial Fractions

Integration using partial fractions is a technique used to integrate rational functions. A rational function is a ratio of two polynomials, say $\frac{N(x)}{D(x)}$. The core idea is to decompose this complex fraction into a sum of simpler fractions whose denominators are the factors of $D(x)$. These simpler fractions are generally easier to integrate.

This calculator specifically handles cases where the denominator $D(x)$ can be factored into a combination of:

• Distinct linear factors (e.g., $(x-a)$)
• Repeated linear factors (e.g., $(x-b)^m$)
• Non-repeated irreducible quadratic factors (e.g., $(ax^2+bx+c)$ where $b^2-4ac < 0$)

The method requires that the degree of the numerator $N(x)$ is strictly less than the degree of the denominator $D(x)$. If this is not the case, polynomial long division must be performed first to express the rational function as a polynomial plus a proper rational function.

Assuming $\frac{N(x)}{D(x)}$ is a proper rational function, the form of its partial fraction decomposition depends on the nature of the factors of $D(x)$.

1. Distinct Linear Factor $(px+q)$

If $(px+q)$ is a non-repeated linear factor of $D(x)$, the decomposition includes a term:

$$ \frac{A}{px+q} $$

The constant $A$ can be found efficiently using the Heaviside cover-up method: $A = \left. \frac{N(x)}{D(x)/(px+q)} \right|_{x = -q/p}$.

Its integral is: $\int \frac{A}{px+q} dx = \frac{A}{p} \ln|px+q|$.

2. Repeated Linear Factor $(px+q)^m$

If $(px+q)^m$ is a factor of $D(x)$ (meaning the linear factor $px+q$ is repeated $m$ times), the decomposition includes a sum of $m$ terms:

$$ \frac{A_1}{px+q} + \frac{A_2}{(px+q)^2} + \cdots + \frac{A_m}{(px+q)^m} $$

The coefficients $A_k$ are found using the formula involving derivatives of $G(x) = \frac{N(x)}{D(x)/(px+q)^m}$ evaluated at the root $x=-q/p$: $A_k = \frac{1}{(m-k)!} G^{(m-k)}(-q/p)$.

The integral of each term $\frac{A_k}{(px+q)^k}$ is:

• For $k=1$: $\frac{A_1}{p} \ln|px+q|$.
• For $k > 1$: $\frac{A_k}{p(1-k)(px+q)^{k-1}}$.

3. Non-Repeated Irreducible Quadratic Factor $(ax^2+bx+c)$

An irreducible quadratic factor is one where $b^2-4ac < 0$, meaning it has no real roots and cannot be factored further into real linear factors. If $(ax^2+bx+c)$ is a non-repeated irreducible quadratic factor of $D(x)$, the decomposition includes a term:

$$ \frac{Ax+B}{ax^2+bx+c} $$

To find the coefficients $A$ and $B$:

1. After determining coefficients for all linear factors, group them on one side: $\frac{N(x)}{D(x)} - \sum (\text{linear partial fractions}) = \frac{Ax+B}{ax^2+bx+c} + \dots$ (other quadratic terms, if any).
2. If this is the only remaining unknown term (or one of a few), multiply both sides by $(ax^2+bx+c)$. The left side becomes a polynomial.
3. $Ax+B = (ax^2+bx+c) \left[ \frac{N(x)}{D(x)} - \sum (\text{known linear terms}) \right]$. The expression on the right should simplify to a linear polynomial.
4. Equate coefficients of $x$ and the constant terms on both sides to solve for $A$ and $B$. For example, if the right side simplifies to $M x + K$, then $A=M$ and $B=K$.

To integrate $\int \frac{Ax+B}{ax^2+bx+c} dx$:

1. Manipulate the numerator: Rewrite $Ax+B$ to relate to the derivative of the denominator. The derivative of $ax^2+bx+c$ is $2ax+b$. So, $Ax+B = \frac{A}{2a}(2ax+b) + \left(B - \frac{Ab}{2a}\right)$. Let $C' = B - \frac{Ab}{2a}$.
2. Split the integral: The integral becomes $\int \frac{\frac{A}{2a}(2ax+b) + C'}{ax^2+bx+c}dx = \frac{A}{2a} \int \frac{2ax+b}{ax^2+bx+c}dx + C' \int \frac{1}{ax^2+bx+c}dx$.
3. First part (logarithm): The first integral is $\frac{A}{2a} \ln|ax^2+bx+c|$, using a u-substitution $u=ax^2+bx+c$.
4. Second part (arctangent): For $C' \int \frac{1}{ax^2+bx+c}dx$, complete the square in the denominator: $ax^2+bx+c = a\left(x^2+\frac{b}{a}x+\frac{c}{a}\right) = a\left(\left(x+\frac{b}{2a}\right)^2 + \frac{c}{a} - \frac{b^2}{4a^2}\right) = a\left(\left(x+\frac{b}{2a}\right)^2 + \frac{4ac-b^2}{4a^2}\right)$. Let $h = \frac{b}{2a}$ and $k^2 = \frac{4ac-b^2}{4a^2}$ (note $k^2 > 0$ since $b^2-4ac < 0$). The integral becomes $\frac{C'}{a} \int \frac{1}{(x+h)^2+k^2}dx$. This evaluates to $\frac{C'}{a} \cdot \frac{1}{k}\arctan\left(\frac{x+h}{k}\right)$.

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

This calculator currently supports:

• Rational functions where the degree of the numerator is strictly less than the degree of the denominator. If not, polynomial long division should be performed manually before using the calculator.
• Denominators that can be factored (by the user) into:
  • Distinct linear factors.
  • Repeated linear factors.
  • Non-repeated irreducible quadratic factors (where $b^2-4ac < 0$).

Currently not supported by the automatic coefficient solver:

• Repeated irreducible quadratic factors (e.g., $(x^2+1)^2$). The general form is known, but coefficient calculation is more involved.
• Simultaneous solving for coefficients of multiple irreducible quadratic factors. The current method focuses on isolating one quadratic term at a time after linear terms are handled. For multiple quadratic factors, a larger system of equations would typically be formed and solved.
• Automatic factorization of the denominator polynomial. The user must provide the denominator in its factored form.