Unit 6.14 – Selecting Techniques for Antidifferentiation

AP® Calculus AB & BC | The Art of Choosing the Right Integration Method

Why This Matters: Integration is both a science and an art! While you've learned many techniques—basic rules, u-substitution, integration by parts (BC), partial fractions (BC), completing the square—the real challenge is knowing which technique to use when. This topic brings it all together, teaching you to recognize patterns, make strategic decisions, and efficiently solve integration problems. Success on AP® exams depends on quickly identifying the best approach. Master this skill and integration becomes much less intimidating!

🧰 Your Integration Toolbox

Complete Integration Techniques

ptionion>All Methods Available

Technique	When to Use	AB/BC
Basic Rules	Direct antiderivatives, polynomials, trig, exp	AB & BC
U-Substitution	Composite functions, chain rule reverse	AB & BC
Algebraic Manipulation	Simplify first, split fractions, factor	AB & BC
Long Division	Improper rational functions	AB & BC
Completing the Square	Quadratics leading to inverse trig	AB & BC
Integration by Parts	Products of different types (LIATE)	BC only
Partial Fractions	Proper rational functions with factored denominators	BC only

🔍 The Decision Process

Master Strategy: Ask These Questions in Order

1️⃣ Can I use BASIC RULES directly?

Is it a simple polynomial, trig, exponential, or power function?
Can I rewrite it as a simple sum of basic functions?
If YES: Integrate directly!
If NO: Continue to next question

2️⃣ Should I SIMPLIFY algebraically first?

Can I expand, factor, or split the expression?
Is it a fraction that can be separated?
Can I cancel terms or simplify radicals?
If YES: Simplify, then reassess
If NO: Continue to next question

3️⃣ Is it a COMPOSITE FUNCTION (u-substitution)?

Do I see \(f(g(x)) \cdot g'(x)\)?
Is there an "inside function" whose derivative appears?
If YES: Use u-substitution!
If NO: Continue to next question

4️⃣ Is it a RATIONAL FUNCTION (fraction)?

If numerator degree ≥ denominator degree: Long division first
If proper and denominator factors: Partial fractions (BC)
If quadratic denominator doesn't factor: Completing the square
If NO: Continue to next question

5️⃣ Is it a PRODUCT of different function types? (BC)

Do I have \(x^n \cdot e^x\), \(x^n \cdot \sin x\), \(\ln x\), etc.?
If YES: Integration by parts (LIATE)
If NO: May need creative approach or combination of methods

🎯 Pattern Recognition Guide

Quick Recognition Table

Spot the Pattern, Choose the Method
You See...	Think...	Method
\(x^n, e^x, \sin x, \cos x\) alone	Direct integration	Basic Rules
\(f(g(x)) \cdot g'(x)\) pattern	Chain rule reverse	U-Substitution
\(\frac{f'(x)}{f(x)}\)	Logarithm!	U-Sub → \(\ln\|f(x)\|\)
\(x \cdot e^x, x \cdot \sin x, \ln x\)	Product of different types	Integration by Parts (BC)
\(\frac{\text{polynomial}}{\text{polynomial}}\)	Check degrees first	Long Division or Partial Fractions (BC)
\(\frac{1}{x^2 + bx + c}\)	Inverse trig coming	Completing the Square
\(\frac{x+a}{(x+b)(x+c)}\)	Factorable denominator	Partial Fractions (BC)
Complicated expression	Simplify first!	Algebraic Manipulation

📖 Decision-Making Examples

Example 1: Quick Recognition

Problem: \(\int 2x(x^2 + 1)^5 \, dx\)

Decision Process:

Basic rules? No, too complex
Simplify? Could expand, but messy
U-substitution? YES! See \((x^2+1)^5\) and its derivative \(2x\)

✓ Method: U-Substitution with \(u = x^2 + 1\)

Solution:

Let \(u = x^2 + 1\), then \(du = 2x\,dx\)

\[ \int u^5 \, du = \frac{u^6}{6} + C = \frac{(x^2+1)^6}{6} + C \]

Example 2: Multiple Possibilities

Problem: \(\int \frac{x^3 + 2x}{x^2 + 1} \, dx\)

Decision Process:

Basic rules? No
Simplify? Maybe split numerator?
U-substitution? Not quite—numerator isn't exactly derivative of denominator
Rational function? YES! Numerator degree (3) ≥ denominator degree (2)

✓ Method: Long Division first, then u-substitution

Solution:

Long division: \(\frac{x^3 + 2x}{x^2 + 1} = x + \frac{x}{x^2+1}\)

\[ \int \left(x + \frac{x}{x^2+1}\right)dx = \frac{x^2}{2} + \frac{1}{2}\ln|x^2+1| + C \]

Example 3: Product Recognition (BC)

Problem: \(\int x e^{2x} \, dx\) (BC only)

Decision Process:

Basic rules? No
U-substitution? No—derivative of \(x\) doesn't match \(e^{2x}\)
Product of different types? YES! \(x\) (algebraic) times \(e^{2x}\) (exponential)

✓ Method: Integration by Parts (LIATE: A before E)

Solution:

Let \(u = x\), \(dv = e^{2x}dx\)

\[ = \frac{xe^{2x}}{2} - \frac{e^{2x}}{4} + C = \frac{e^{2x}(2x-1)}{4} + C \]

Example 4: Simplify First!

Problem: \(\int \frac{\sin^2 x}{\cos^2 x} \, dx\)

Decision Process:

Recognize? This is \(\tan^2 x\)!
Simplify! Use identity: \(\tan^2 x = \sec^2 x - 1\)

✓ Method: Algebraic Manipulation → Basic Rules

Solution:

\[ \int (\sec^2 x - 1)\,dx = \tan x - x + C \]

Example 5: Multiple Steps

Problem: \(\int \frac{5x + 3}{x^2 + 4x + 13} \, dx\)

Decision Process:

Rational function? Yes, proper fraction
Does denominator factor? Check discriminant: \(16 - 52 < 0\) → No!
Strategy: Split into two integrals—one for ln, one for arctan

✓ Method: Rewrite numerator + Completing the Square

Solution:

Rewrite: \(5x + 3 = \frac{5}{2}(2x + 4) - 7\)

Complete square: \(x^2 + 4x + 13 = (x+2)^2 + 9\)

\[ = \frac{5}{2}\ln|x^2+4x+13| - \frac{7}{3}\arctan\left(\frac{x+2}{3}\right) + C \]

❌ Common Selection Mistakes

Mistake 1: Jumping to complex methods when basic rules work
Mistake 2: Not simplifying algebraically first
Mistake 3: Missing u-substitution opportunities
Mistake 4: Forgetting to check if rational function is improper
Mistake 5: Using integration by parts when u-substitution works
Mistake 6: Not recognizing \(\frac{f'(x)}{f(x)} = \ln|f(x)|\) pattern
Mistake 7: Trying u-substitution when parts is needed (BC)
Mistake 8: Missing opportunities to split fractions
Mistake 9: Not completing the square when denominator doesn't factor
Mistake 10: Giving up too quickly—try multiple approaches!

💡 Master Strategies

✅ General Principles:

Start simple: Try basic rules and simplification first
Look for patterns: Composite functions, products, quotients
LIATE for products (BC): Logarithmic, Inverse trig, Algebraic, Trig, Exponential
When stuck: Try simplifying algebraically
Practice recognition: The more you do, the faster you get
Check your answer: Differentiate to verify!

🔥 Quick Checks:

See \(x^n f(x)\)? Try parts (BC) with \(u = x^n\)
See \(\frac{g'(x)}{g(x)}\)? Answer is \(\ln|g(x)|\)
See \(e^{kx}\) or \(\sin(kx)\)? Watch for constant adjustments
Numerator = derivative of denominator? Logarithm!
Perfect derivative match? U-substitution for sure

📝 Practice: Identify the Method

For each integral, identify the best method(s):

\(\int 3x^2 e^{x^3} \, dx\)
\(\int \frac{x}{x^2 + 1} \, dx\)
\(\int \frac{x^2 + 3x + 5}{x + 2} \, dx\)
\(\int x \cos x \, dx\) (BC)
\(\int \frac{1}{x^2 + 6x + 13} \, dx\)
\(\int (x + 1)^2 \, dx\)
\(\int \frac{2x + 1}{x^2 - 4} \, dx\)
\(\int \sin^2 x \cos x \, dx\)

Answers:

U-Substitution: \(u = x^3\), \(du = 3x^2\,dx\) → \(\frac{e^{x^3}}{3} + C\)
U-Substitution: \(u = x^2+1\) → \(\frac{1}{2}\ln(x^2+1) + C\)
Long Division first: Then integrate → \(\frac{x^2}{2} + x + 3\ln|x+2| + C\)
Integration by Parts (BC): \(u = x\), \(dv = \cos x\,dx\) → \(x\sin x + \cos x + C\)
Completing the Square: \((x+3)^2 + 4\) → \(\frac{1}{2}\arctan(\frac{x+3}{2}) + C\)
Expand first (Simplify): → \(\frac{x^3}{3} + x^2 + x + C\)
Partial Fractions (BC): Factor denominator first
U-Substitution: \(u = \sin x\) → \(\frac{\sin^3 x}{3} + C\)

✏️ AP® Exam Success Strategies

What Makes You Successful on Exams:

Speed comes from recognition: Practice identifying patterns
Don't waste time on wrong methods: If stuck after 30 seconds, try different approach
Show your work: Even if method doesn't work out, you may get partial credit
Check reasonableness: Does answer make sense?
For BC: Know when to use parts vs. substitution
Time management: Skip and return if integral seems too difficult
Verify when possible: Quick differentiation check

💯 Exam Day Checklist:

Read problem completely—what type of integral?
Check for immediate patterns (basic rules, u-sub)
Consider if simplification helps
For rational functions: proper/improper?
For products: integration by parts? (BC)
If stuck: try algebraic manipulation
Write clearly—show method selection
Don't forget +C for indefinite integrals!

⚡ Ultimate Quick Reference

THE DECISION FLOWCHART

Step 1: Can you integrate directly?

→ YES: Use basic rules

→ NO: Go to Step 2

Step 2: Should you simplify first?

→ YES: Expand, factor, split, then reassess

→ NO: Go to Step 3

Step 3: Is it composite (inside function + derivative)?

→ YES: U-Substitution

→ NO: Go to Step 4

Step 4: Is it a rational function?

→ Improper: Long Division

→ Proper + factors: Partial Fractions (BC)

→ Quadratic doesn't factor: Complete Square

→ NO: Go to Step 5

Step 5 (BC): Is it a product of different types?

→ YES: Integration by Parts (LIATE)

→ NO: Try creative approaches or combination

Master the Art of Selection! Success in integration comes from pattern recognition. Start with the simplest approach: Can you use basic rules? If not, should you simplify algebraically first? Next, check for u-substitution opportunities—look for composite functions where the derivative of the inside appears. For rational functions, check if improper (long division), proper with factorable denominator (partial fractions-BC), or unfactorable quadratic (completing square). For products of different function types (BC), use integration by parts with LIATE priority. Remember key patterns: \(\frac{f'(x)}{f(x)} = \ln|f(x)|\), \(f(g(x)) \cdot g'(x)\) screams u-substitution, \(x^n e^x\) or \(x^n \sin x\) suggests parts (BC). Practice until recognition becomes automatic—the method should jump out at you! When stuck, try simplifying or splitting the integral. Always verify by differentiating your answer. Speed comes from experience, so practice extensively with varied problems. Master this skill and you'll tackle any integral with confidence! 🎯✨