Choosing an integration technique is a pattern recognition skill that turns a difficult antiderivative into a manageable plan. Many integrals do not simplify by basic rules alone, so students need a way to decide what structure to look for first. A decision flowchart helps by connecting the form of the integrand to a method such as substitution, integration by parts, partial fractions, or trigonometric substitution.
This matters because a good first choice often saves several pages of algebra and prevents circular work.
Start by asking what form the integral has: a composite function, a product of different types of functions, a rational expression, or a radical involving a quadratic. Substitution reverses the chain rule, integration by parts reverses the product rule, partial fractions breaks rational functions into simpler pieces, and trigonometric substitution converts certain radicals into trig identities. Good technique choice also depends on algebraic preparation, such as factoring denominators, completing the square, or rewriting powers.
The best integrators test the structure before calculating and switch methods when the chosen path does not simplify the integral.
Key Facts
- Substitution is best when the integrand contains a function and a constant multiple of its derivative: ∫ f(g(x))g'(x) dx = ∫ f(u) du.
- Integration by parts is useful for products where differentiating one factor simplifies it: ∫ u dv = uv - ∫ v du.
- Partial fractions applies to rational functions P(x)/Q(x) when deg(P) < deg(Q) or after long division.
- For rational functions, factor the denominator first: linear factors give terms like A/(x - a), and repeated factors give A1/(x - a) + A2/(x - a)^2.
- Trig substitution is often used for radicals: a^2 - x^2 suggests x = a sin θ, a^2 + x^2 suggests x = a tan θ, and x^2 - a^2 suggests x = a sec θ.
- Always include the constant of integration for indefinite integrals: ∫ f(x) dx = F(x) + C.
Vocabulary
- Substitution
- A technique that replaces an inner expression with a new variable to reverse the chain rule.
- Integration by parts
- A technique based on the product rule that rewrites the integral of a product as a simpler integral.
- Partial fractions
- A method for rewriting a rational expression as a sum of simpler rational expressions.
- Trigonometric substitution
- A technique that uses trigonometric identities to simplify radicals involving quadratic expressions.
- Rational function
- A function that can be written as a ratio of two polynomials, P(x)/Q(x).
Common Mistakes to Avoid
- Using substitution when no derivative match appears: this is wrong because u-substitution only helps when dx can be converted cleanly into du.
- Choosing integration by parts with a poor u: this is wrong because if u does not simplify when differentiated, the new integral may become harder than the original.
- Applying partial fractions before checking the degree: this is wrong because an improper rational function must be divided first when deg(P) ≥ deg(Q).
- Using trig substitution without matching the radical form: this is wrong because each substitution depends on a specific identity, such as 1 - sin^2 θ = cos^2 θ or 1 + tan^2 θ = sec^2 θ.
Practice Questions
- 1 Evaluate ∫ 2x cos(x^2 + 1) dx and identify the technique used.
- 2 Evaluate ∫ x e^(3x) dx using integration by parts.
- 3 For each integral, choose the most appropriate first technique and explain why: ∫ (3x^2)/(x^3 + 5) dx, ∫ x ln x dx, ∫ dx/(x^2 - 4), and ∫ sqrt(9 - x^2) dx.