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Find the integral of (x² - 4)^(3/2)

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Final answer:

To find the integral of (x² - 4)^(3/2), use a substitution by letting u = x² - 4. Then, integrate with respect to u to get the final answer.

Step-by-step explanation:

To find the integral of (x² - 4)^(3/2), we can use a substitution. Let u = x² - 4. Then, du/dx = 2x, and dx = du/(2x). Substitute these into the integral to get:

∫(x² - 4)^(3/2) dx = ∫u^(3/2) √(2/x) du = ∫ 2^(3/2) √x^(-3/2) u^(3/2) du

Now, integrate with respect to u:

∫ 2^(3/2) √x^(-3/2) u^(3/2) du = 2^(3/2) √x^(-3/2) ∫ u^(3/2) du = 2^(3/2) √x^(-3/2) * (2/5)u^(5/2) + C = (4/5) √x^(-3/2) (x² - 4)^(5/2) + C. This result showcases the integral of the given expression, providing a means to compute and understand complex integrals through strategic substitution and integration methods

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