Completing The Square Integral Calculator

What is Completing the Square?

Completing the square is an algebraic method used to rewrite quadratic expressions in the form of a perfect square trinomial. This technique is particularly useful in calculus when dealing with integrals that contain quadratic expressions in their denominators.

The general form of a quadratic expression is:

By completing the square, we can rewrite this expression as:

where d and e are constants determined by the coefficients of the original quadratic expression.

How to Complete the Square

To complete the square for a quadratic expression ax² + bx + c, follow these steps:

1. Factor out the coefficient of x² from the first two terms: a(x² + (b/a)x) + c.
2. Take half of the coefficient of x, square it, and add and subtract this value inside the parentheses: a(x² + (b/a)x + (b/2a)² - (b/2a)²) + c.
3. Rewrite the expression as a perfect square trinomial plus the remaining terms: a[(x + b/2a)² - (b/2a)²] + c.
4. Distribute the a and combine like terms to get the final expression in the form a(x + d)² + e.

Completing the Square Integral

When integrating functions that contain quadratic expressions in their denominators, completing the square can simplify the integral significantly. The general form of such an integral is:

By completing the square for the quadratic expression in the denominator, we can rewrite the integral in a more manageable form. The steps for completing the square integral are as follows:

1. Identify the quadratic expression in the denominator and complete the square as described in the previous section.
2. Rewrite the integral using the completed square form of the denominator.
3. Use substitution to simplify the integral, typically by letting u = x + d, where d is the constant from the completed square form.
4. Integrate the resulting expression and back-substitute to obtain the final result.

Example Calculation

Let's consider the integral:

To solve this integral using completing the square, follow these steps:

1. First, complete the square for the denominator x² + 4x + 5:
   x² + 4x + 5 = (x² + 4x + 4) + 1 = (x + 2)² + 1
2. Rewrite the integral using the completed square form:
   ∫ (2x + 3)/[(x + 2)² + 1] dx
3. Use substitution by letting u = x + 2, which implies du = dx:
   ∫ (2(u - 2) + 3)/(u² + 1) du = ∫ (2u - 4 + 3)/(u² + 1) du = ∫ (2u - 1)/(u² + 1) du
4. Split the integral into two parts and integrate separately:
   ∫ (2u)/(u² + 1) du - ∫ 1/(u² + 1) du
5. Evaluate each integral:
   ∫ (2u)/(u² + 1) du = ln(u² + 1) ∫ 1/(u² + 1) du = arctan(u)
6. Combine the results and back-substitute u = x + 2:
   ln((x + 2)² + 1) - arctan(x + 2) + C

The final result of the integral is: