Integration Techniques Summary

Calculus 2 · Axiom Academy

SUMMARY Integration Techniques Summary The complete toolkit of integration methods — what each technique is, when to reach for it, and how they combine. Pattern recognition is the skill. Unlike differentiation, integration has no universal algorithm — success comes from recognizing which technique fits the form in front of you. Six core methods cover almost everything: u-substitution, integration by parts, trigonometric integrals, trigonometric substitution, partial fractions, and numerical approximation. Hard integrals combine techniques. A single problem may need a substitution after parts, or a trig identity inside a trig substitution — break it into manageable steps. Numerical methods bridge the gap. When no elementary antiderivative exists (like ), the Trapezoidal and Simpson's rules deliver accurate approximations. Always verify. Check any indefinite integral by differentiating the result. Reverses the chain rule. Let u = g(x) and , then rewrite the entire integral in terms of u . When to use: a composition f(g(x)) where the inner function's derivative appears (or can be manufactured) in the integrand. Watch out for: definite integrals — either change the limits to u -values or back-substitute before evaluating. Technique Integration by Parts Reverses the product rule. Split the integrand into a part to differentiate ( u ) and a part to integrate ( dv ). When to use: products of functions from different families, e.g. , x^2 e^ x , .

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