Integration by Completing the Square
Calculate integrals of rational functions using completing the square and substitution methods
Calculate Integration by Completing the Square
Cannot be zero
Positive integer (1-5)
Integration Result
Solution Steps:
Step 1: Original Function
Step 2: Complete the Square
Step 3: Substitution
Step 4: Simplified Integral
Step 5: Apply Integration Formula
Analysis
Example Calculation
Example: ∫ 1/(x² + 4x + 13) dx
Given: a = 1, b = 4, c = 13
Step 1: Complete the square: x² + 4x + 13 = (x + 2)² + 9
Step 2: Substitute u = x + 2, du = dx
Step 3: ∫ 1/(u² + 9) du = (1/3) × arctan(u/3) + C
Result: (1/3) × arctan((x + 2)/3) + C
Integration Methods
Completing the Square
For irreducible quadratics
Substitution
u = x + h transformation
Standard Forms
Apply known integral formulas
Common Integral Forms
Understanding Integration by Completing the Square
What is Completing the Square?
Completing the square is a method used to rewrite quadratic expressions in the form a(x + h)² + k. This technique is particularly useful for integrating rational functions where the denominator is an irreducible quadratic.
When to Use This Method?
- •The denominator is a quadratic expression
- •The quadratic cannot be factored easily
- •The discriminant is negative (no real roots)
- •Partial fraction decomposition is complex
Step-by-Step Process
Step 1: Factor out coefficient
If a ≠ 1, factor it out from the quadratic
Step 2: Complete the square
Add and subtract (b/2a)² to form a perfect square
Step 3: Substitute
Let u = x + h to simplify the expression
Step 4: Apply formula
Use standard integral formulas for the result