Evaluate the integral.

Question

\int_0^(.75) (x^(2))/(\sqrt(9-16x^(2)))dx

Answer 1

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Answer 2

I = ∫0.75x²/(9 - 16x²)^1/2dx; x = 3/4sinθ, dx = 3/4cosθdθ; I = ∫(3/4)⁴sin²θcosθdθ/(3cosθ) = 27/512∫(1 - cos2θ)dθ = 27/512(θ - sin2θ/2) + C; θ = sin^-1(4x/3); sin2θ/2 = sinθcosθ = 4x/3(1 - 16x²/9)^1/2 = 4x(9 - 16x²)^1/2.

So, I = 27/512[sin^-1(4x/3) - 4x(9 - 16x²)^1/2] + C