Integration by Completing the Square and Substitution

When an integral contains a quadratic expression of the form ax² + bx + c in the denominator or under a square root, standard anti-derivative formulas do not directly apply. By normalizing the leading coefficient 'a', we find the linear term coefficient b1 = b/a and the constant term c1 = c/a. We then complete the square by taking half of the linear coefficient, squaring it (b2_sq = (b1/2)²), and adding or subtracting it to keep the expression balanced. This transforms the original trinomial into a sum or difference of squares, such as (x + b3)² ± d3, which easily yields to standard integration formulas involving inverse trigonometric functions like arctangent or arcsine.

Let us walk through evaluating an integral featuring the quadratic denominator x² + 4x + 7. First, identify our coefficients where a = 1, b = 4, and c = 7. We compute the linear term adjustment: b1 = 4 / 1 = 4. Next, we find half of this value and square it: b2 = 4 / 2 = 2, so b2_sq = 2² = 4. To complete the square, we rewrite x² + 4x + 7 as (x² + 4x + 4) + 3, which simplifies to the squared binomial form (x + 2)² + 3. Here, our discriminant or constant shift d3 equals 3. This clean transformation allows a direct substitution u = x + 2, converting the integral into a standard form that evaluates smoothly using inverse tangent rules.

Always factor out the leading coefficient 'a' from the x² and x terms before attempting to complete the square; forgetting this step is the most common source of arithmetic errors. Pay close attention to signs when dealing with negative coefficients to ensure your substituted binomial remains balanced. Finally, always verify your final integrated expression by taking its derivative using the chain rule to confirm it matches the original integrand.