6.14 Selecting Techniques for Antidifferentiation

Building a strategy toolkit

By 6.14 you should combine u-substitution, integration by parts, partial fractions, and known antiderivative forms to tackle any integral. First simplify algebraically (expand, factor, split fractions) before picking a method. Look for a function-derivative pair (substitution), a product of unlike function types (parts), or a rational function with factorable denominator (partial fractions).

Recognizing the right cue

Substitution works when you spot g(x) and g'(x) together. Integration by parts suits products like x·eˣ or x·ln(x) — pick u to differentiate nicely (LIATE order). Partial fractions applies to proper rational functions with factorable denominators. Trig identities may simplify integrals of trig powers before integrating.

Check your work

After antidifferentiating, differentiate your answer to verify it matches the original integrand — this catches sign errors and missed constants. Also remember the +C for indefinite integrals, and simplify the final expression when possible for clarity.

Key terms

When to use u-substitution

When the integrand contains a function and its derivative (g(x) and g'(x)) together.

When to use integration by parts

When integrating a product of two different types of functions (e.g., polynomial × exponential/log/trig).

When to use partial fractions

For proper rational functions with a denominator that factors into linear or irreducible quadratic terms.

LIATE

Order for choosing 'u' in integration by parts: Logarithmic, Inverse trig, Algebraic, Trig, Exponential.

How to verify an antiderivative

Differentiate your result — it should equal the original integrand.

Try an AP-style question

∫ x cos(x²) dx = ?

1. 2 sin(x²) + C
2. (x²/2) sin x + C
3. (1/2) sin(x²) + C
4. sin(x²) + C