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.
- 2 worked examples
- 16 in Techniques of Integration
Worked examples
- 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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Common questions
What is the 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. It is one of the techniques of integration formulas in the CalcRef reference.
How do you use the Trig Integrals: sinᵐx cosⁿx?
Worked example. 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.
Practice
Quiz yourself on Techniques of Integration
Trig Integrals: sinᵐx cosⁿx is one of 16 Techniques of Integration formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.