Strategies for Trig Integrals

Calculus 2 · Axiom Academy

LESSON Strategies for Trig Integrals A systematic way to integrate products of sine and cosine — decided entirely by whether their powers are odd or even. Almost every problem in this family fits a single shape: a power of times a power of . Everything hinges on the two exponents m and n . One question splits the whole family in two. Ask it first, and the technique chooses itself. Strip off one , convert the rest with , then let . No factor to strip. Use power reduction: and . You have a choice — strip off whichever factor is more convenient and follow Case 1 or Case 2. When sine carries an odd power, peel off a single and turn the remaining even power into cosines: Sine has the odd power (3), so use this case. 4. Case 2: Odd Power of Cosine When cosine carries the odd power, the process mirrors Case 1 — swap the roles of sine and cosine: Cosine has the odd power (5), so use this case. With both powers even there is no single factor to peel off, so u -substitution stalls. Instead, reach for the power-reduction formulas: Both powers are even, so use power reduction. Apply power reduction again to . Simplify and integrate term by term. You now have a single decision to make on any product of sines and cosines — read the powers, then follow the branch. Scroll up to revisit any step.

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