6.14 Selecting Antidifferentiation Techniques

Match the method

Recognize the form of the integrand: use the power rule for simple powers, u-substitution for a function-and-its-derivative pattern, and algebraic rewriting (long division, completing the square, splitting fractions) when needed first.

Keep it simple

If a basic rule applies, use it rather than a more complex method. Efficiency comes from identifying the structure before integrating.

Key terms

Selecting a technique

Match the method to the integrand's form.

Power rule

For simple powers of x.

u-substitution

For a function-and-its-derivative pattern.

Algebraic rewriting

Long division / completing the square when needed.

Efficiency

Use the simplest rule that applies.

Try an AP-style question

∫ x cos(x²) dx = ?

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