#Integration Using Completing the Square

#Table of Contents

What is Completing the Square?
Integrating Using Completing the Square
Worked Example
Practice Questions
Glossary
Summary and Key Takeaways

#What is Completing the Square?

#Introduction

Completing the square is a method used to rewrite quadratic expressions in a way that makes them easier to manipulate, particularly for solving equations and integrating.

#Quadratic Expression

A quadratic expression is of the form: $a{x}^{2} + bx + c$

#Simple Version of Completing the Square

For a quadratic expression without a leading coefficient: ${x}^{2} + bx + c = \left(x + \frac{b}{2}\right)^{2} + c - \left(\frac{b}{2}\right)^{2}$

Example:

{x}^{2} - 6x + 2 = \left(x - 3\right)^{2} + 2 - \left(-3\right)^{2} = \left(x - 3\right)^{2} - 7

#Completing the Square with a Leading Coefficient

When a coefficient is in front of the $x^{2}$ term: $a{x}^{2} + bx + c = a\left(x + \frac{b}{2a}\right)^{2} + c - a\left(\frac{b}{2a}\right)^{2}$

Example:

3{x}^{2} + 12x - 7 = 3\left(x + 2\right)^{2} - 7 - 3\left(2\right)^{2} = 3\left(x + 2\right)^{2} - 19

#Integrating Using Completing the Square

#Standard Integrals

Recall the following standard integrals: $\int \frac{1}{\sqrt{1 - x^{2}}} , dx = \mathrm{arcsin}x + C, \quad -1 < x < 1$ $\int \frac{1}{1 + x^{2}} , dx = \mathrm{arctan}x + C$

#Using Completing the Square in Integration

Completing the square can recast integrals into forms where the above results can be applied. Sometimes, methods from the Integrals of Composite Functions study guide may also be needed.

Example:

\int \frac{1}{x^{2} - 6x + 13} \, dx

Complete the square on the denominator: $x^{2} - 6x + 13 = \left(x - 3\right)^{2} + 4$
Rewrite the function to be integrated: $\frac{1}{x^{2} - 6x + 13} = \frac{1}{4 + \left(x - 3\right)^{2}} = \frac{1}{4} \cdot \frac{1}{1 + \left(\frac{x - 3}{2}\right)^{2}}$
Recognize the standard form: This is now a multiple of $\frac{1}{1 + (\cdots)^{2}}$ . Hence, the integral will involve $\mathrm{arctan}(\cdots)$ .
Differentiate $\mathrm{arctan}\left(\frac{x - 3}{2}\right)$ to check: $\frac{d}{dx}\left(\mathrm{arctan}\left(\frac{x - 3}{2}\right)\right) = \frac{1}{1 + \left(\frac{x - 3}{2}\right)^{2}} \cdot \frac{1}{2} = \frac{1}{2} \cdot \frac{1}{1 + \left(\frac{x - 3}{2}\right)^{2}}$
Integrate: $\int \frac{1}{x^{2} - 6x + 13} , dx = \frac{1}{4} \int \frac{1}{1 + \left(\frac{x - 3}{2}\right)^{2}} , dx = \frac{1}{2} \mathrm{arctan}\left(\frac{x - 3}{2}\right) + C$

#Worked Example

#Problem

Find the indefinite integral: $\int \frac{1}{\sqrt{21 - x^{2} + 4x}} , dx$

#Solution

Complete the square on the expression inside the square root: $21 - x^{2} + 4x = -\left(x - 2\right)^{2} + 25$
Rewrite the function to be integrated: $\frac{1}{\sqrt{21 - x^{2} + 4x}} = \frac{1}{\sqrt{25 - \left(x - 2\right)^{2}}} = \frac{1}{5} \cdot \frac{1}{\sqrt{1 - \left(\frac{x - 2}{5}\right)^{2}}}$
Recognize the standard form: This is now a multiple of $\frac{1}{\sqrt{1 - (\cdots)^{2}}}$ . Hence, the integral will involve $\mathrm{arcsin}(\cdots)$ .
Differentiate $\mathrm{arcsin}\left(\frac{x - 2}{5}\right)$ to check: $\frac{d}{dx}\left(\mathrm{arcsin}\left(\frac{x - 2}{5}\right)\right) = \frac{1}{\sqrt{1 - \left(\frac{x - 2}{5}\right)^{2}}} \cdot \frac{1}{5}$
Integrate: $\int \frac{1}{\sqrt{21 - x^{2} + 4x}} , dx = \int \frac{1}{5} \cdot \frac{1}{\sqrt{1 - \left(\frac{x - 2}{5}\right)^{2}}} , dx = \mathrm{arcsin}\left(\frac{x - 2}{5}\right) + C$

Thus, $\int \frac{1}{\sqrt{21 - x^{2} + 4x}} , dx = \mathrm{arcsin}\left(\frac{x - 2}{5}\right) + C$

#Practice Questions

Practice Question

Integrate: $\int \frac{1}{x^{2} + 4x + 5} , dx$

Practice Question

Integrate: $\int \frac{1}{\sqrt{9 - 6x + x^{2}}} , dx$

#Glossary

Quadratic Expression: A polynomial of degree 2, generally written as $ax^2 + bx + c$ .
Completing the Square: A method used to rewrite a quadratic expression as a perfect square trinomial plus a constant.
Arcsin: The inverse sine function.
Arctan: The inverse tangent function.

#Summary and Key Takeaways

#Summary

Completing the square simplifies quadratic expressions and is a powerful tool in integration.
Standard integrals involving $\mathrm{arcsin}$ and $\mathrm{arctan}$ can be used once the quadratic expression is transformed into the appropriate form.

#Key Takeaways

Always convert the quadratic expression to a perfect square form.
Use standard integrals to find the antiderivative after completing the square.
Check your solution by differentiating to ensure accuracy.

Exam Tip

Completing the square is especially useful for integrals involving quadratic polynomials. Master this technique to handle a variety of integration problems effectively.

Common Mistake

Don't forget to adjust and compensate after completing the square, especially when dealing with coefficients in front of $x^2$ .