Exclusive to AA HL, trigonometric integrals apply identities and substitutions to integrate powers and products of trigonometric functions. Students evaluate integrals of powers of sine and cosine — for example ∫sin³x cos²x dx — using the Pythagorean identity sin²x + cos²x = 1 and, for even powers, the double-angle identities; they integrate powers of tangent and secant, such as ∫tan³x sec³x dx, using the identity 1 + tan²x = sec²x. The section also covers trigonometric substitution, where substituting x = a sinθ, x = a tanθ or x = a secθ transforms integrands containing √(a² − x²), √(a² + x²) or √(x² − a²) into trigonometric integrals that standard identities then simplify.
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