Problem 1 - Calculus Early Transcendentals

Question

What change of variables is suggested by an integral containing \(\sqrt{x^{2}-9} ?\)

Step-by-Step Solution

Answer

#Answer# The change of variables suggested is \(x = 3\cosh(u)\).

Step 1: Identify the substitution

In order to simplify the expression \(\sqrt{x^2 - 9}\), we can use the hyperbolic sine function \(\sinh(u)\) because of its identity \(\cosh^2(u) - 1 = \sinh^2(u)\). The change of variables that can be suggested is \(x = 3\cosh(u)\). This substitution is chosen because when \(x=3\cosh(u)\), the expression becomes \(\sqrt{(3\cosh(u))^2 - 9}\), which will simplify to \(3\sinh(u)\) that is easier to work with.

Step 2: Differentiate the substitution

We need to find the differential of \(x\) with respect to \(u\), or \(\frac{dx}{du}\), to make the substitution in the integral. Differentiate \(x = 3\cosh(u)\) with respect to \(u\): \(\frac{dx}{du} = 3\sinh(u)\).

Step 3: Perform the change of variables

Now we have all the pieces needed to perform the change of variables in the given integral. If we let \(I\) represent the integral containing \(\sqrt{x^2 - 9}\), performing this substitution would yield a new integral with respect to \(u\). For example, if the integral was given as \(\int{\sqrt{x^2 - 9}} dx\), substituting the expression for \(x\) and \(\frac{dx}{du}\), we get the new integral: \(I = \int{3\sinh(u) * (3\sinh(u)) du}\). After performing the change of variables, we can proceed to solve the new integral, simplifying the expression and finding the antiderivative in terms of \(u\). Finally, to evaluate the original integral for \(x\), replace \(u\) with the inverse hyperbolic function, \(u = \cosh^{-1}(\frac{x}{3})\).