Integration Strategy
This section doesn't introduce new math — it teaches metacognition. How do you look at a random integral and decide which technique to use?
The Decision Flowchart
When faced with an integral, work through these steps in order:
Simplify First
Can you use algebra or trig identities to rewrite the integral?
- • Expand polynomials:
- • Simplify fractions:
- • Use trig identities:
Look for Substitution
Is the derivative of an inner function present (or close to present)?
If you see , try .
Classify the Form
Product of different types?
Integration by Parts
Rational function?
Partial Fractions
Powers of sin/cos?
Trig Identities
Square roots of quadratics?
Trig Substitution
Try Again
If nothing works, manipulate the integral and repeat the process.
- • Multiply by clever forms of 1 (conjugates)
- • Add and subtract terms
- • Complete the square
- • Long division if degree of numerator ≥ denominator
Pattern Recognition
Learn to recognize these common patterns and immediately know which method to use:
Algebraic × Transcendental
, ,
→ Integration by Parts
Polynomial / Polynomial
,
→ Partial Fractions
Powers of Sin/Cos
,
→ Trig Identities + u-sub
, ,
Square roots of sums/differences of squares
→ Trig Substitution
nth roots of linear expressions
→ Rationalizing Substitution
in denominator
Unfactorable quadratic
→ Complete the Square
Pro Tip
Always check for hidden substitutions first! An integral that looks complicated might become trivial with the right .
Strategy Examples
Example 1: Hidden Substitution
Evaluate .
This looks scary! But wait... the derivative of is , and there's a in the denominator!
Let , so :
This is now a simple power rule integral (derivative of the inside is in the numerator):
Example 2: Multiply by Conjugate
Evaluate .
No immediate substitution. No familiar form. Manipulation needed!
Using :
Example 3: Long Division First
Evaluate .
Degree of numerator (3) > degree of denominator (2). Must do long division first!
Now the integral becomes:
First integral is power rule. Second is u-sub with :
Practice Quiz
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