Evaluate the integral ∫-4/3(-2√2)/3 (3/x√(9x2-4))dx

Question

Evaluate the integral ∫_-4/3^(-2√2)/3 (3/x√(9x²-4))dx. Please show all steps.

Answer 1

Substitution x = 2/3secu; dx = 2/3secutanudu; 2/3secu = - 4/3; secu = - 2; u = 2π/3; 2/3secu = - 2√2/3; secu =

-√2; u = 3π/4∫

∫_-4/3^(-2√2)/3 (3/x√(9x²- 4))dx = ∫_2π/3^3π/4·9/2cosu·2tanudu = - 9cosu_2π/3^3π/4= - 9(cos3π/4 - cos2π/3) = - 9(-√2/2 + 1/2) = 9(√2 - 1)/2