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Chapter 11: Problem 6

Evaluate the indefinite integral. $$ \int \frac{d x}{\sqrt{x^{2}-a^{2}}} $$

Short Answer

Expert verified
\(\text{cosh}^{-1}\left(\frac{x}{a}\right) + C\)

Step by step solution

01

Recognize the Integral Form

Identify that the integral \(\frac{1}{\sqrt{x^2 - a^2}}\) fits the standard integral form for the inverse hyperbolic cosine function.
02

Recall Standard Integral

Recall that the integral for \(\frac{1}{\sqrt{x^2 - a^2}}\) is \(\text{cosh}^{-1}\left(\frac{x}{a}\right) + C\), where C is the constant of integration.
03

Apply the Formula

Apply the formula to solve the problem: \(\text{cosh}^{-1}\left(\frac{x}{a}\right) + C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Hyperbolic Functions
In mathematics, inverse hyperbolic functions are functions that reverse the effect of hyperbolic functions like sinh, cosh, and tanh. These functions are denoted as arcsinh, arccosh, and arctanh, respectively. In our exercise, we recognize that the integral \(\frac{1}{\text{sqrt}(x^2 - a^2)}\) fits the inverse hyperbolic cosine form. This means we need to use the inverse hyperbolic cosine function (arccosh) to solve it. The formula for the integral \(\frac{1}{\text{sqrt}(x^2 - a^2)}\) is \(\text{cosh}^{-1}(\frac{x}{a}) + C\).
Understanding a bit about hyperbolic functions can be beneficial:
  • **Cosh(x)**: The hyperbolic cosine function, which is defined as \(\text{cosh}(x) = \frac{e^x + e^{-x}}{2}\)
  • **Inverse Cosh(x)**: The inverse hyperbolic cosine function (arccosh or cosh^{-1}(x)) reverses the effect of cosh. So, \(\text{cosh}^{-1}(x)\) finds the value whose hyperbolic cosine is x.
Standard Integral Forms
When solving indefinite integrals, it's useful to recognize standard integral forms. These forms provide a shortcut to quickly identify and integrate functions. In this exercise, our function \(\frac{1}{\text{sqrt}(x^2 - a^2)}\) matches the standard integral for the inverse hyperbolic cosine function.
Some common standard integral forms include:
  • \(\frac{1}{\text{sqrt}(x^2 - a^2)}\) giving \(\text{cosh}^{-1}(\frac{x}{a}) + C\)
  • \(\frac{1}{\text{sqrt}(a^2 - x^2)}\) giving \(\text{sin}^{-1}(\frac{x}{a}) + C\)
  • \(\frac{1}{x^2 + a^2}\) giving \(\frac{1}{a}\text{tan}^{-1}(\frac{x}{a}) + C\)
Recognizing these forms simplifies solving integrals and reinforces understanding of different types of functions and their behaviors.
Constant of Integration
When dealing with indefinite integrals, we always include a constant of integration. This constant, denoted as \(C\), represents an arbitrary constant added to the indefinite integral solution. Why do we need this constant?
  • **Completeness**: Every indefinite integral represents a family of functions. Adding C ensures this family is complete.
  • **Unique Solutions**: In applications such as initial value problems, C helps us find unique solutions fitting specific conditions.
For our exercise, the solution is \(\text{cosh}^{-1}(\frac{x}{a}) + C\). Adding C means we do not miss any possible solutions from the family of functions that can satisfy the integral process.

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