Find the indefinite integral using the substitution x = 2 sec(𝜃). (Use C for the constant of integration.) ∫x^3√x^2 − 4 dx

Answer 1

x = 2secθ; dx = 2secθtanθdθ; Then ∫x^3√x^2 − 4 dx

= ∫8sec³θ·2tanθ·2secθtanθdθ = 32 ∫sec⁴θtan²θdθ = 32 ∫(tan² + 1)tan²θd(tanθ) = 32(tan⁵θ/5 + tan³θ/3) + C;

tanθ = √x² - 4/2 So, our integral I = (√x² - 4)⁵/5 + 4(√x² - 4)³/3 + C

Answer 2

Here is a Desmos graph showing one form for the antiderivative as well as a table that indicates it is likely the correct answer:

desmos.com/calculator/upe91ordp6