Three Key Substitutions

Calculus 2 · Axiom Academy

LESSON Three Key Substitutions When a square root blocks an integral, the right trig substitution turns it into an identity that erases the radical. 1. Three Patterns, Three Identities Every trig substitution rests on a rearranged Pythagorean identity. Match the form under the root to the identity that turns it into a perfect square — then the radical disappears. Play the map to see all three at once. Use this for . Setting makes , so the radical becomes . Watch the reference triangle build and the radical leg collapse. Why it works: , so — on we have . Use this for . Setting makes , so the radical becomes . This time the radical lives on the hypotenuse . Use this for . Setting makes , so the radical becomes . Now x is the hypotenuse and the radical is the opposite leg. Why it works: , so on the chosen domain. 5. Choosing the Right Substitution The whole method hinges on pattern recognition: read the form under the root, and the sign and order pick the substitution for you. Play the sorter to route each radical to its method. Three radicals, three substitutions, three Pythagorean identities — each one chosen to make the square root vanish. Scroll up to revisit any substitution.

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