Topic 6.10 Notes – Integrating Functions Using Long Division and Completing the Square

Rewriting the Integrand So It Matches a Known Antiderivative

Sometimes an integral looks impossible because it isn’t in a recognizable form yet. The goal is always the same:

Rearrange the expression so it fits a standard antiderivative.

Common target forms you already know:

• $\int x^n \, dx$
• $\int \frac{1}{x} dx = \ln|x| + C$
• $\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C$
• $\int \frac{1}{x^2 + a^2} dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C$
• $\int \frac{1}{\sqrt{a^2 - x^2}} dx = \arcsin\left(\frac{x}{a}\right) + C$

If the structure doesn’t match one of these, fix the algebra first.

Two tools make that happen:

• Polynomial long division
• Completing the square

Integrating Rational Functions Using Long Division

You’re dealing with a rational function when you have a polynomial divided by a polynomial.

When to Use It

Use long division when:

• The numerator’s degree is greater than or equal to the denominator’s degree.

If the numerator’s degree is smaller, skip long division.

What the Rewrite Accomplishes

Long division splits the integrand into:

• A polynomial
• Plus a proper fraction (top degree < bottom degree)

That makes the integral manageable.

Example

$$\int \frac{x^2 + 3x + 5}{x + 2} \, dx$$

Since degree 2 ≥ degree 1, divide first.

After long division:

$$\frac{x^2 + 3x + 5}{x + 2} = x + 1 + \frac{3}{x + 2}$$

Now integrate term by term:

$$\int \left(x + 1 + \frac{3}{x+2}\right) dx$$

$$= \frac{1}{2}x^2 + x + 3\ln|x+2| + C$$

Notice what happened:

• Polynomial → power rule
• Proper fraction → natural log

On multiple choice, if answers look like “polynomial + ln term,” that’s a big hint long division was required.

Definite Integrals

Same process:

1. Divide.
2. Integrate.
3. Apply bounds.

No $+C$. Watch algebra carefully when plugging in limits.

Completing the Square

This shows up when a quadratic expression needs to match an inverse trig form.

Typical signals:

• Denominator contains a quadratic that doesn’t factor nicely.
• Expression under a square root looks close to $a^2 - x^2$.
• You suspect an arctan or arcsin answer.

The Algebra Step

Take something like:

$$x^2 - 6x + 13$$

Complete the square:

$$x^2 - 6x + 13 = (x^2 - 6x + 9) + 4 = (x - 3)^2 + 4$$

Now it matches the form $(x-h)^2 + a^2$.

Matching Inverse Trig Forms

Here are the shapes you’re trying to create:

Arctangent Form

If you get:

$$\int \frac{1}{(x - h)^2 + a^2} dx$$

Use:

$$\frac{1}{a}\arctan\left(\frac{x - h}{a}\right) + C$$

You may need to factor out constants first.

Example

$$\int \frac{2}{x^2 - 4x + 8} dx$$

Complete the square:

$$x^2 - 4x + 8 = (x - 2)^2 + 4$$

Rewrite:

$$2\int \frac{1}{(x-2)^2 + 2^2} dx$$

Apply formula:

$$2 \cdot \frac{1}{2}\arctan\left(\frac{x-2}{2}\right) + C$$

$$= \arctan\left(\frac{x-2}{2}\right) + C$$

Arcsine Form

If you get:

$$\int \frac{1}{\sqrt{a^2 - (x - h)^2}} dx$$

Use:

$$\arcsin\left(\frac{x - h}{a}\right) + C$$

Be careful with signs. If the quadratic starts negative, factor out −1 first.

Choosing the Right Strategy

Quick mental checklist:

• Rational function with top degree ≥ bottom → long division
• Quadratic inside denominator or radical → complete the square
• After rewriting, check for:
  • $\ln|f(x)|$
  • arctan form
  • arcsin form

This topic is heavily algebra-driven. Most mistakes come from small algebra errors, not calculus.