Techniques of Integration · Trigonometric Methods
Trig Integrals: secᵐx tanⁿx
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.
- 2 worked examples
- 16 in Techniques of Integration
Worked examples
- m = 4 is even. Save sec²x: ∫ sec²x tan²x sec²x dx
- sec²x = 1+tan²x: ∫ (1+tan²x) tan²x sec²x dx
- u = tanx, du = sec²x dx: ∫ (1+u²)u² du = ∫ (u²+u⁴) du
- = u³/3 + u⁵/5 + C = tan³x/3 + tan⁵x/5 + C
Answer: tan³x/3 + tan⁵x/5 + C
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Common questions
What is the Trig Integrals: secᵐx tanⁿx?
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. It is one of the techniques of integration formulas in the CalcRef reference.
How do you use the Trig Integrals: secᵐx tanⁿx?
Worked example. Find ∫ sec⁴x tan²x dx. m = 4 is even. Save sec²x: ∫ sec²x tan²x sec²x dx. sec²x = 1+tan²x: ∫ (1+tan²x) tan²x sec²x dx. u = tanx, du = sec²x dx: ∫ (1+u²)u² du = ∫ (u²+u⁴) du. = u³/3 + u⁵/5 + C = tan³x/3 + tan⁵x/5 + C. Answer: tan³x/3 + tan⁵x/5 + C.
Practice
Quiz yourself on Techniques of Integration
Trig Integrals: secᵐx tanⁿx 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.