Integrate By Completing The Square

Integrating by Completing the Square: A Comprehensive Guide

Integrating functions can be challenging, especially when dealing with complex expressions in the denominator. One powerful technique to simplify these integrals and make them solvable is completing the square. This method is particularly useful when dealing with quadratic expressions within the integrand, allowing us to transform the integral into a form readily integrable using standard techniques. This article will provide a comprehensive guide to integrating by completing the square, covering the theoretical underpinnings, step-by-step procedures, diverse examples, and frequently asked questions.

Understanding Completing the Square

Before diving into integration, let's refresh our understanding of completing the square. This algebraic technique transforms a quadratic expression of the form ax² + bx + c into a perfect square trinomial plus a constant. The general process involves:

1. Ensure the coefficient of x² is 1: If it's not, factor out the coefficient from the x² and x terms.

2. Find half of the coefficient of x: Take half of the coefficient of the x term (b/2).

3. Square the result: Square the value obtained in step 2 ((b/2)²).

4. Add and subtract the squared value: Add and subtract this squared value inside the parentheses.

5. Rewrite as a perfect square: Rewrite the expression as a perfect square trinomial (x + b/2)² plus or minus the remaining constant.

Example: Complete the square for the quadratic expression x² + 6x + 5.

1. The coefficient of x² is already 1.

2. Half of the coefficient of x (6) is 3.

3. Squaring 3 gives 9.

4. Adding and subtracting 9: x² + 6x + 9 - 9 + 5

5. Rewriting as a perfect square: (x + 3)² - 4

Therefore, x² + 6x + 5 can be rewritten as (x + 3)² - 4.

Integrating Using Completing the Square: A Step-by-Step Approach

The application of completing the square in integration primarily involves transforming the denominator of a rational function into a perfect square plus or minus a constant. This allows us to use standard integration formulas or techniques like substitution to solve the integral.

Steps:

1. Identify the appropriate integral: Look for integrals where the denominator is a quadratic expression.

2. Complete the square in the denominator: Apply the completing the square technique to the quadratic expression in the denominator.

3. Rewrite the integral: Rewrite the integral with the completed square in the denominator.

4. Use an appropriate integration technique: Depending on the form of the integral, use a suitable technique such as substitution, trigonometric substitution, or partial fraction decomposition. Frequently, a simple u-substitution will suffice.

5. Evaluate the integral: Evaluate the resulting integral using standard integration formulas.

6. Add the constant of integration: Remember to always add the constant of integration, '+ C', to your final answer.

Illustrative Examples

Let's work through several examples to solidify our understanding.

Example 1: Evaluate ∫ dx / (x² + 4x + 8)

1. Complete the square: The denominator is x² + 4x + 8. Completing the square, we get (x + 2)² + 4.

2. Rewrite the integral: The integral becomes ∫ dx / ((x + 2)² + 4).

3. Substitution: Let u = x + 2, then du = dx. The integral transforms to ∫ du / (u² + 4).

4. Standard integral: This is a standard integral of the form ∫ du / (u² + a²), which evaluates to (1/a)arctan(u/a) + C. In our case, a = 2.

5. Evaluate and substitute back: The integral evaluates to (1/2)arctan(u/2) + C. Substituting back u = x + 2, we get (1/2)arctan((x + 2)/2) + C.

Example 2: Evaluate ∫ dx / (x² - 6x + 5)

1. Complete the square: Completing the square for x² - 6x + 5 gives (x - 3)² - 4.

2. Rewrite the integral: The integral becomes ∫ dx / ((x - 3)² - 4).

3. Substitution: Let u = x - 3, then du = dx. The integral becomes ∫ du / (u² - 4).

4. Partial Fraction Decomposition: We can factor the denominator as (u - 2)(u + 2) and use partial fraction decomposition to break the integral into simpler parts: A/(u - 2) + B/(u + 2). Solving for A and B gives A = 1/4 and B = -1/4.

5. Integrate and substitute back: The integral becomes (1/4)∫ du/(u - 2) - (1/4)∫ du/(u + 2), which evaluates to (1/4)ln|u - 2| - (1/4)ln|u + 2| + C. Substituting back u = x - 3, we get (1/4)ln|x - 5| - (1/4)ln|x - 1| + C, which can be simplified to (1/4)ln|(x - 5)/(x - 1)| + C.

Example 3: Evaluate ∫ (2x + 3) dx / (x² + 2x + 5)

1. Complete the square: The denominator is x² + 2x + 5, which completes to (x + 1)² + 4.

2. Rewrite the integral: The integral becomes ∫ (2x + 3) dx / ((x + 1)² + 4).

3. Substitution: This time, we'll use a slightly more sophisticated substitution. Let u = x + 1, then du = dx and x = u - 1. Substituting these into the integral, we get ∫ (2(u - 1) + 3) du / (u² + 4) = ∫ (2u + 1) du / (u² + 4).

4. Splitting the integral: We can split this into two simpler integrals: ∫ 2u du / (u² + 4) + ∫ du / (u² + 4).

5. Integrate: The first integral can be solved using a simple u-substitution (let v = u² + 4). The second integral is a standard arctangent integral.

6. Combine and substitute back: After integrating and substituting back for u and x, we get ln|x² + 2x + 5| + (1/2)arctan((x + 1)/2) + C.

Advanced Considerations and Variations

While the examples above showcase the common applications, completing the square can be adapted for more intricate integrals. For instance, integrals involving square roots of quadratic expressions may also benefit from this technique, often leading to trigonometric substitutions after completing the square.

Frequently Asked Questions (FAQ)

Q1: What if the coefficient of x² is not 1?

A1: Factor out the coefficient of x² from the x² and x terms before completing the square. Remember to adjust the integral accordingly.

Q2: Can completing the square be used with cubic or higher-order polynomials?

A2: No, completing the square is specifically designed for quadratic expressions. For higher-order polynomials, different techniques like partial fraction decomposition or other advanced methods are typically required.

Q3: What if completing the square results in a negative constant inside the square?

A3: This often leads to integrals involving hyperbolic functions or requires further manipulation.

Q4: Why is completing the square useful in integration?

A4: It simplifies the integrand, transforming it into a form that is easily integrable using standard techniques or substitutions. This makes complex-looking integrals much more manageable.

Conclusion

Integrating by completing the square is a valuable tool in the calculus arsenal. It provides an elegant and efficient way to simplify integrals with quadratic expressions in the denominator. By mastering this technique, you'll significantly expand your ability to solve a wider range of integration problems and gain a deeper understanding of integral calculus. Remember the step-by-step procedure, practice with diverse examples, and don’t hesitate to consult additional resources for further exploration and refinement of your skills. Consistent practice is key to mastering this crucial technique.