Integrating Functions Using Long Division and Completing the Square

Integrating Functions Using Long Division and Completing the Square: AP Calculus Study Guide

Ever encountered an integral that made you want to cry into your calculator? Fear not! Today, we’re diving into the magical realms of polynomial long division and completing the square, two secret weapons you can deploy when basic techniques leave you stumped. All those algebra and pre-calc days will finally pay off! 🙌

Let’s start with long division, the algebraic equivalent of cutting your food into bite-sized chunks when it’s too big to chew in one go.

Imagine you’re faced with a rational function, something like a polynomial sandwich where both the numerator and denominator are polynomials. If the numerator’s degree is as big or bigger than the denominator’s, long division is your go-to power move.

Here’s an example of how to handle it:

Example Walkthrough: 📝

Suppose we have: [ \int \frac{2x^2 - 4}{x + 1} , dx ]

First things first, spot the rational function! The numerator is a quadratic (degree 2) and the denominator is linear (degree 1). Time to whip out polynomial long division:

1. Divide ( 2x^2 - 4 ) by ( x + 1 ).

   Using long division, we get: [ 2x - 2 \quad \text{with a remainder of} \quad -\frac{2}{x+1} ]

2. Rewrite the integral with the result of the division: [ \int (2x - 2 - \frac{2}{x+1}) , dx ]

3. Now we can integrate each term separately: [ \int 2x , dx - \int 2 , dx - 2 \int \frac{1}{x+1} , dx ]

   Which becomes: [ x^2 - 2x - 2 \ln |x + 1| + C ]

Bravo! You’ve handled long division with the grace of a mathematician! Also, keep an eye on multiple-choice questions—if answers look like polynomials with natural log terms, you might be looking at a long-division problem.

Good news! Completing the square isn’t just useful for putting quadratic equations into vertex form—it’s also handy for integrating tricky expressions. Think of it as transforming a gnarly quadratic into a form that’s much easier to digest.

Here’s how completing the square helps:

Example Walkthrough: 📐

Consider the integral: [ \int \frac{4}{t^2 - 4t + 20} , dt ]

1. First, complete the square in the denominator. This means turning ( t^2 - 4t + 20 ) into something friendlier: [ t^2 - 4t + 20 = (t^2 - 4t + 4) + 16 = (t - 2)^2 + 16 ]

2. Rewrite the integral: [ \int \frac{4}{(t - 2)^2 + 16} , dt ]

3. Notice anything? This denominator looks suspiciously like the form for an arctan integral. To simplify, let's factor out the 4 in the denominator: [ 4 \int \frac{1}{(t - 2)^2 + 4^2} , dt ]

4. Now use the arctan integral formula: [ \int \frac{1}{x^2 + a^2} , dx = \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C ]

   Here, ( x = t - 2 ) and ( a = 4 ):

   [ 4 \cdot \frac{1}{4} \arctan \left( \frac{t - 2}{4} \right) + C ]

5. Simplify: [ \arctan \left( \frac{t - 2}{4} \right) + C ]

Amazing work! You manipulated, squared, and tanned (not in the sunburn sense) that integral into submission!

Here’s where you get to channel your inner calculus wizard:

Practice 1: Evaluate: [ \int \frac{2x^3 + 3x^2 - 17x - 27}{x^2 - 9} , dx ]

1. Use polynomial long division.
2. Integrate the result.

Practice 2: Evaluate: [ \int \frac{1}{\sqrt{3 - x^2 - 2x}} , dx ]

1. Simplify the quadratic in the integrand by completing the square.
2. Integrate by recognizing the form of the integrand.

Practice 3: Evaluate: [ \int \frac{x^2}{x+1} , dx ]

1. Use polynomial long division as necessary.
2. Integrate each term from the division.

You’ve unlocked some powerful tools in your calculus arsenal! Long division and completing the square can simplify integrals that seem impossible at first glance. Practice makes perfect, so keep at it, and soon these techniques will be second nature!

Happy integrating, calculus champions! May your equations always balance and your denominators never get too pesky. 📏🔢