CalcRef

Techniques of Integration · Trigonometric Methods

Trig Integrals: secᵐx tanⁿx

∫sec⁡mxtan⁡nx dx\int \sec^m x \tan^n x \, dx

Strategy: If n is odd, save sec x tan x and convert tan² = sec²-1. If m is even, save sec²x and convert sec² = 1+tan².

Worked examples

Find ∫ sec⁴x tan²x dx.
  1. m = 4 is even. Save sec²x: ∫ sec²x tan²x sec²x dx
  2. sec²x = 1+tan²x: ∫ (1+tan²x) tan²x sec²x dx
  3. u = tanx, du = sec²x dx: ∫ (1+u²)u² du = ∫ (u²+u⁴) du
  4. = u³/3 + u⁵/5 + C = tan³x/3 + tan⁵x/5 + C

Answer: tan³x/3 + tan⁵x/5 + C

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