Evaluate the integral. (Use C for the constant of integration.)

Question

\int (dx)/((x^(2)+4)^((5)/(2)))

Answer 1

If you combine the two fractions that Yefim derived, you will get

(1/24)x(x²+6)/(x²+4)^3/2 + C

Which might match the required answer.

Answer 2

θI = ∫dx/(x² + 4)^5/2; x = 2tanθ; dx = 2sec²θdθ; I = ∫2sec²θdθ/(32sec⁵θ) = 1/16∫cos³θdθ = 1/16∫(1 - sin²θ)d(sinθ) = 1/16(sinθ - sin³θ/3) + C

sinθ = tanθcosθ = x/2/(1 + x²/4)^1/2 = x/(x² + 4)^1/2.

I = 1/16[x/(x² + 4)^1/2 + 1/3x³/(x² + 4)^3/2] + C