6.14 - Selecting Techniques for Antidifferentiation

Overview

This lesson is about recognizing which integration technique to apply. On exams, problems won’t tell you which method to use. You have to identify it yourself.

Integration Techniques Summary

For AB and BC Students

1. Power Rule (Basic Antiderivatives)

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

When to use: polynomials, and terms with integer or fractional exponents.

2. U-Substitution

\int f(g(x)) \cdot g'(x) \, dx

When to use: composite functions where the derivative of the inside function shows up (or is close to showing up).

Key: look for a function and its derivative both sitting in the integrand.

3. Trigonometric Integrals

\int \sin x \, dx = -\cos x + C

\int \cos x \, dx = \sin x + C

\int \sec^2 x \, dx = \tan x + C

When to use: basic trig functions appear in the integrand.

4. Inverse Trigonometric Patterns

\int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1}(x) + C

\int \frac{1}{1+x^2} \, dx = \tan^{-1}(x) + C

When to use: you recognize the specific patterns with

\sqrt{1-x^2}

1+x^2

, and so on.

5. Exponential and Logarithmic

\int e^x \, dx = e^x + C

\int \frac{1}{x} \, dx = \ln|x| + C

When to use:

e^x

\frac{1}{x}

appears, though you may need u-sub for variations.

6. Algebraic Manipulation

Polynomial long division: when the numerator’s degree is greater than or equal to the denominator’s

Completing the square: to create inverse trig patterns or simplify radicals

Algebraic simplification: separate fractions, distribute products

When to use: the expression doesn’t match any standard form and needs rewriting first.

Additional Techniques (BC Only)

7. Integration by Parts

\int u \, dv = uv - \int v \, du

When to use: a product of functions where one gets simpler when differentiated. Use LIATE.

8. Partial Fractions

Break rational functions into simpler fractions.

When to use: a rational function with a factorable denominator.

9. Improper Integrals

\int_a^\infty f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx

When to use: infinite limits, or vertical asymptotes in the interval.

The Challenge: Recognition

On exams:

There are no hints about which method to use

You have to analyze the integrand and identify the technique

You may need to combine several techniques

Strategy:

Look for patterns. Does it match a known form?

Check for composition. Is u-substitution possible?

Examine the structure. Do you need polynomial division? Completing the square?

Consider the domain. Any discontinuities? Infinite limits?

Think about products. Would integration by parts help? (BC)

Decision-Making Flowchart

When you see an integral, ask:

Is it a basic form? Use the power rule, basic trig, exponential, or log

Is it a product of a function and its derivative? U-substitution

Does it match an inverse trig pattern? Recognize it and apply directly

Is the numerator’s degree greater than or equal to the denominator’s? Long division first

Do you need to create a standard form? Complete the square

Is it a product of different function types? (BC) Integration by parts

Are there infinite limits or vertical asymptotes? (BC) Improper integral, so use limits

Is it rational with a factorable denominator? (BC) Partial fractions

Practice Mindset

What makes this hard isn’t the techniques themselves. It’s knowing which one to apply when.

Success requires:

A solid understanding of each individual technique

Pattern recognition skills

Experience spotting when each method applies

A willingness to try more than one approach

Exam reality: problem #17 isn’t going to say “use long division.” You have to see it yourself.

Practice: Name the Technique

Try each one before reading the last two columns. Watch for pairs that look almost the same but need different techniques.

Integral	What you notice	Technique	Answer
$\displaystyle\int \dfrac{x}{x^2 + 9} \, dx$	The top is almost the derivative of the bottom	U-substitution, $u = x^2 + 9$	$\dfrac{1}{2}\ln(x^2 + 9) + C$
$\displaystyle\int \dfrac{1}{x^2 + 9} \, dx$	No $x$ on top, a sum of squares on the bottom	Inverse tangent (factor out the 9)	$\dfrac{1}{3}\tan^{-1}\left(\dfrac{x}{3}\right) + C$
$\displaystyle\int \dfrac{x^2 + 1}{x + 1} \, dx$	The top’s degree is at least the bottom’s	Long division	$\dfrac{x^2}{2} - x + 2\ln\lvert x + 1 \rvert + C$
$\displaystyle\int \dfrac{1}{x^2 + 4x + 5} \, dx$	A quadratic on the bottom that doesn’t factor	Complete the square	$\tan^{-1}(x + 2) + C$
$\displaystyle\int x e^{x^2} \, dx$	$x^2$ inside, and its derivative (almost) outside	U-substitution, $u = x^2$	$\dfrac{1}{2}e^{x^2} + C$
$\displaystyle\int \tan x \, dx$	Rewrite it as $\dfrac{\sin x}{\cos x}$	U-substitution, $u = \cos x$	$-\ln\lvert \cos x \rvert + C$
$\displaystyle\int x e^{x} \, dx$	A product with no derivative to cancel	(BC) Integration by parts, $f = x$	$xe^x - e^x + C$
$\displaystyle\int \dfrac{1}{x^2 - 4} \, dx$	The bottom factors into different linear factors	(BC) Partial fractions	$\dfrac{1}{4}\ln\left\lvert \dfrac{x - 2}{x + 2} \right\rvert + C$
$\displaystyle\int_0^1 \dfrac{1}{\sqrt{x}} \, dx$	A vertical asymptote at $x = 0$	(BC) Improper integral	$2$ (converges)

Key Takeaways

Recognition is the skill. Identifying which technique to use is the actual challenge

No hints on exams. You have to analyze the integrand on your own

Multiple techniques. You’ll often need to combine methods, like completing the square then u-sub

AB vs BC: know which techniques are required for your exam level

Practice variety. Seeing lots of different problems builds recognition

Think strategically. Analyze the structure before diving into calculations

Review previous lessons. Each technique was taught separately, and now you integrate them

This is synthesis. It brings together everything from Unit 6

Check your toolkit. Make sure you’re comfortable with each individual technique

Unit 6 is complete, and Unit 7 is next

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