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Chapter 7: Problem 1

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

Short Answer

Expert verified
The appropriate change of variable suggests the substitution \(x = 3\cosh(\theta)\) with the corresponding differential change \(dx = 3\sinh(\theta)d\theta\), and the simplified expression is \(\sqrt{x^2 - 9} = 3\sinh(\theta)\). This substitution helps to simplify the integral and make it easier to compute.

Step by step solution

01

Identify the type of substitution

Given the expression \(\sqrt{x^2-9}\), we can notice that it is similar to the trigonometric identity \(x^2 - a^2 = (x-a)(x+a)\), where \(a=3\). This suggests using a suitable trigonometric substitution.
02

Choose the appropriate trigonometric substitution

We can try the hyperbolic trigonometric function, \(\cosh(\theta)\), with the substitution: $$ x=3\cosh(\theta) $$ Here, \(\theta\) is the variable we will substitute for \(x\).
03

Differentiate x with respect to 胃

Now we differentiate \(x = 3\cosh(\theta)\) with respect to \(\theta\) to get the differential change: $$ \frac{dx}{d\theta} = 3\sinh(\theta) $$ As a result, the differential change is \(dx = 3\sinh(\theta)d\theta\).
04

Substitute and simplify the expression

Let's substitute \(x=3\cosh(\theta)\) into the expression and simplify: $$ \sqrt{x^2 - 9} = \sqrt{(3\cosh(\theta))^2-9} $$ Here, we notice that \((3\cosh(\theta))^2-9 = 9(\cosh^2(\theta)-1)\), and the trigonometric identity for hyperbolic cosine states that \(\cosh^2(\theta)-1 = \sinh^2(\theta)\). Therefore, we can write: $$ \sqrt{x^2 - 9} = \sqrt{9\sinh^2(\theta)} = 3\sinh(\theta) $$
05

Conclusion

The appropriate change of variable for an integral containing the expression \(\sqrt{x^2 - 9}\) suggests the following substitution: $$ x = 3\cosh(\theta) $$ with the corresponding differential change \(dx = 3\sinh(\theta)d\theta\), and the simplified expression: $$ \sqrt{x^2 - 9} = 3\sinh(\theta) $$ This substitution will help to simplify the integral and make it easier to compute.

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