Techniques of Integration · Trigonometric Methods
Trig Integrals: sinᵐx cosⁿx
Strategy: If m or n is odd, save one factor and convert the rest using sin²+cos²=1. If both even, use half-angle identities: sin²x = (1-cos2x)/2, cos²x = (1+cos2x)/2.
- 0 variables
- 2 worked examples
- 16 in Techniques of Integration
Worked examples
Find ∫ sin³x cos²x dx.
- m = 3 is odd. Save one sinx: ∫ sin²x cos²x sinx dx
- sin²x = 1-cos²x: ∫ (1-cos²x) cos²x sinx dx
- u = cosx, du = -sinx dx: -∫ (1-u²)u² du = -∫ (u²-u⁴) du
- = -u³/3 + u⁵/5 + C = -cos³x/3 + cos⁵x/5 + C
Answer: -cos³x/3 + cos⁵x/5 + C
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Find ∫ sin²x dx.
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Practice
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