CalcRef

Techniques of Integration · Trigonometric Methods

Trig Integrals: sinᵐx cosⁿx

∫sin⁡mxcos⁡nx dx\int \sin^m x \cos^n x \, dx

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.

Worked examples

Find ∫ sin³x cos²x dx.
  1. m = 3 is odd. Save one sinx: ∫ sin²x cos²x sinx dx
  2. sin²x = 1-cos²x: ∫ (1-cos²x) cos²x sinx dx
  3. u = cosx, du = -sinx dx: -∫ (1-u²)u² du = -∫ (u²-u⁴) du
  4. = -u³/3 + u⁵/5 + C = -cos³x/3 + cos⁵x/5 + C

Answer: -cos³x/3 + cos⁵x/5 + C

Find ∫ sin²x dx.
  1. Both even. Use half-angle: sin²x = (1-cos2x)/2
  2. (1/2)∫ (1-cos2x) dx = x/2 - sin2x/4 + C

Answer: x/2 - sin(2x)/4 + C

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