Evaluating ∫sin³(x)cos²(x)dx

Calculus 2 · Axiom Academy

Use an odd power of sine and a u -substitution to evaluate a trigonometric integral. Evaluate the indefinite integral . The power of sine is 3 (odd) and the power of cosine is 2 (even). Nice work — you evaluated a trigonometric integral with the odd-power technique. Here's what carried the solution: Spot the odd power: when sine or cosine appears to an odd power, peel off one factor to pair with du in the substitution. Convert with the Pythagorean identity: rewrite the remaining even power using so everything is in the substitution variable. Choose : it works because the leftover is exactly -du . Mind the negative: since , factor the minus sign out front before integrating. Expand, then integrate: distribute to , integrate each term with the power rule, and back-substitute. This strategy handles any where m or n is odd — peel off one factor of the odd-power function, convert the rest, and substitute.

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