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What change of variables is suggested by an integral containing √(x² - 144)

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Final answer:

To evaluate the integral √(x² - 144), it is suggested to use a change of variables. Let x = 12sinh(t) and find dx in terms of dt. Substitute x and dx in the integral with the new variables and simplify the integral to evaluate it.

Step-by-step explanation:

To evaluate an integral containing √(x² - 144), it is suggested to use a change of variables. In this case, we can let x = 12sinh(t). This change of variables simplifies the integral and allows us to solve it more easily.

First, let's find dx in terms of dt:

dx = 12cosh(t)dt

Next, let's express x² - 144 in terms of t:

x² - 144 = (12sinh(t))² - 144 = 144sinh²(t) - 144 = 144(sinh²(t) - 1)

Now, substitute x and dx in the integral with the new variables:

∫ √(x² - 144) dx = ∫ √(144(sinh²(t) - 1)) 12cosh(t) dt

Once this substitution is made, you can simplify the integral and evaluate it accordingly.

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