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Rational Function Integration via Completing the Square

When integrating rational functions, one often encounters quadratic denominators that resist simple antiderivatives. The completing square substitution—also known as complete the square integration—transforms expressions like $\int \frac{dx}{ax^2+bx+c}$ into a standard form that yields inverse trigonometric functions or can be tackled via reduction formulas.

Preliminary Steps

Before applying the completing‑square technique, examine the degrees of the numerator and denominator:

• If the numerator’s degree equals or exceeds the denominator’s degree, perform polynomial long division. The quotient is a polynomial (easily integrated with the power rule), and the remainder becomes a proper rational function.
• Express that proper rational function as a sum of partial fractions. For each irreducible quadratic factor $x^2+bx+c$ (or, more generally, $ax^2+bx+c$), the corresponding term in the partial‑fraction decomposition is $\frac{Ax+B}{(x^2+bx+c)^n}$. The method of completing the square handles precisely these denominators.

Completing the Square Substitution for $\frac{1}{(ax^2+bx+c)^n}$

Consider the integral

$$I = \int \frac{dx}{(ax^2+bx+c)^n},$$

where the quadratic does not factor over the reals. The procedure consists of three actions:

1. Complete the square in the denominator:

   $$ax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right).$$

   Denote the constant term $\Delta = c - \frac{b^2}{4a}$. For the simplest case $a=1$, this reduces to $(x+\frac{b}{2})^2 + \Delta$.

2. Perform a linear substitution that eliminates the linear term. Set

   $$u = \frac{x + \frac{b}{2a}}{\sqrt{\Delta/a}}, \qquad \text{or, when } a=1,\; u = \frac{x+\frac{b}{2}}{\sqrt{\Delta}}.$$

   Concretely, when $a=1$, let $x + \frac{b}{2} = \sqrt{\Delta}\,u$; then $dx = \sqrt{\Delta}\,du$. Substituting yields

   $$I = \int \frac{\sqrt{\Delta}\,du}{\bigl( (\sqrt{\Delta}\,u)^2 + \Delta \bigr)^n} = \int \frac{\sqrt{\Delta}\,du}{(\Delta u^2 + \Delta)^n} = \frac{1}{\Delta^{\,n-\frac{1}{2}}} \int \frac{du}{(1+u^2)^n}.$$

3. Evaluate the reduced integral $\int \frac{du}{(1+u^2)^n}$.

   • For $n=1$, the result is $\arctan u + C$.
   • For $n>1$, integration‑by‑parts or reduction formulas from an integral table are required.

Practical Remarks

• The same substitution works for any quadratic coefficient $a$ after a slight re‑scaling; the final expression always involves $\arctan$ (for $n=1$) or more complex rational‑trigonometric forms.
• A quadratic integral calculator typically automates these steps, but understanding the algebra behind the completing square integration makes it easier to handle variations and to verify results.
• Mastering this method is a key milestone in rational function integration, and practicing a few examples will help internalise the procedure.

In summary, the completing square substitution turns a seemingly intractable integral of $1/(ax^2+bx+c)^n$ into a manageable form, leveraging the well‑known integral of $1/(1+u^2)^n$. Whether computed by hand or with the aid of a calculator, this technique is indispensable in calculus.