Evaluate the Trig Integral Using A Trig Substitution Method

Question

Evaluate the Trig Integral Using A Trig Substitution Method. Please show all steps. If possible please attach a video going through the problem.

Answer 1

x = 1/2sinu; dx = 1/2cosudu; x = 1/4; sinu = 1/2; u = π/6; x = √3/4; sinu = √3/2; u = π/3

∫_1/4^√3/4√1 - 4x²dx = ∫_π/6^π/3cosu·1/2cosudu = ∫_π/6^π/31/4(1 + cos2u)du = (1/4u + 1/8sin2u)_π/6^π/3 =

(π/12 +√3/16) - (π/24 + √3/16) = π/24

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