Chapter 2: Problem 42
Use tables to perform the integration. $$ \int \frac{d x}{\sqrt{x^{2}+16}} $$
Short Answer
Expert verified
\( \sinh^{-1}\left(\frac{x}{4}\right) + C \)
Step by step solution
01
Identify the Integration Formula
The given integral \( \int \frac{d x}{\sqrt{x^{2}+16}} \) resembles the standard integration formula for \( \int \frac{d x}{\sqrt{x^2 + a^2}} \). Recognize that this falls under the category of inverse hyperbolic sine functions.
02
Determine the Appropriate Integral Table Formula
From the standard integration table, the integral of \( \frac{1}{\sqrt{x^2 + a^2}} \) is \( \sinh^{-1}\left(\frac{x}{a}\right) + C \). Comparing \( x^2 + 16 \) to the standard \( x^2 + a^2 \), we identify \( a = 4 \).
03
Apply the Standard Formula
Substitute \( a = 4 \) into the standard integration result, giving us: \( \sinh^{-1}\left(\frac{x}{4}\right) + C \). This applies directly as the solution to the integral \( \int \frac{d x}{\sqrt{x^{2}+16}} \).
04
Write Down the Result
The integral \( \int \frac{d x}{\sqrt{x^{2}+16}} \) evaluates to \( \sinh^{-1}\left(\frac{x}{4}\right) + C \), where \( C \) is the constant of integration.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inverse Hyperbolic Functions
Inverse hyperbolic functions, such as the inverse hyperbolic sine (\( \sinh^{-1} \)), are an essential part of calculus, especially when dealing with integrals involving square roots of quadratic expressions. These functions are analogous to their trigonometric counterparts but are based on hyperbolic geometry.
Inverse hyperbolic functions can be represented in terms of logarithmic functions, which helps in analytically solving integrals. For instance, the inverse hyperbolic sine of a variable, \( \sinh^{-1}(x) \), can be expressed as \( \ln(x + \sqrt{x^2 + 1}) \). This dual nature of representation allows for flexibility in integration.
In practical scenarios, these functions simplify the integration process by providing a direct form that corresponds to a common integral. They are particularly useful in solving integrals where they naturally appear in solutions, as in the problem involving \( \int \frac{d x}{\sqrt{x^2+16}} \). By recognizing this fit, one can immediately apply the known formula for inverse hyperbolic functions to find the integral.
Inverse hyperbolic functions can be represented in terms of logarithmic functions, which helps in analytically solving integrals. For instance, the inverse hyperbolic sine of a variable, \( \sinh^{-1}(x) \), can be expressed as \( \ln(x + \sqrt{x^2 + 1}) \). This dual nature of representation allows for flexibility in integration.
In practical scenarios, these functions simplify the integration process by providing a direct form that corresponds to a common integral. They are particularly useful in solving integrals where they naturally appear in solutions, as in the problem involving \( \int \frac{d x}{\sqrt{x^2+16}} \). By recognizing this fit, one can immediately apply the known formula for inverse hyperbolic functions to find the integral.
Standard Integration Tables
Standard integration tables are invaluable resources in calculus, providing a list of integrals for commonly encountered functions. These tables are designed to assist students and mathematicians to swiftly find integrals without performing the integration process from scratch each time.
They list formulas for integrals of both simple and complex functions, identified through their mnemonic structures. For example, the integral of \( \frac{1}{\sqrt{x^2 + a^2}} \) is associated with the inverse hyperbolic sine and can be found directly as \( \sinh^{-1}\left(\frac{x}{a}\right) + C \) in these tables.
By using integration tables, one can easily match an integral to its corresponding formula, making the solution process quicker and reducing the chance of errors. This is especially helpful when dealing with integrals that may not appear straightforward at first glance but reveal their structure upon closer examination.
They list formulas for integrals of both simple and complex functions, identified through their mnemonic structures. For example, the integral of \( \frac{1}{\sqrt{x^2 + a^2}} \) is associated with the inverse hyperbolic sine and can be found directly as \( \sinh^{-1}\left(\frac{x}{a}\right) + C \) in these tables.
By using integration tables, one can easily match an integral to its corresponding formula, making the solution process quicker and reducing the chance of errors. This is especially helpful when dealing with integrals that may not appear straightforward at first glance but reveal their structure upon closer examination.
Integration Techniques
Integration techniques include a variety of methods used to find an integral of a function. These methods range from basic formulas to sophisticated techniques involving transformations and substitutions.
When dealing with the integral \( \int \frac{d x}{\sqrt{x^2+16}} \), recognizing the form as a standard inverse hyperbolic function allows us to utilize substitution method if not using the table. We identify a variable substitution or use an identity to simplify the expression, aligning it to a known form.
Some primary integration techniques include:
When dealing with the integral \( \int \frac{d x}{\sqrt{x^2+16}} \), recognizing the form as a standard inverse hyperbolic function allows us to utilize substitution method if not using the table. We identify a variable substitution or use an identity to simplify the expression, aligning it to a known form.
Some primary integration techniques include:
- Substitution: changing the variable to simplify the integral.
- Partial Fraction Decomposition: breaking down complex rational expressions.
- Integration by Parts: useful for products of functions.
Definite and Indefinite Integrals
Integrals can be classified into two main types: definite and indefinite. An indefinite integral, such as \( \int \frac{d x}{\sqrt{x^{2}+16}} \), represents a family of functions and includes an arbitrary constant of integration, denoted as \( C \). This type of integral does not specify any limits and denotes a general solution.
On the other hand, a definite integral defines the integration over a specific interval and provides a single numerical value rather than a function. Definite integrals are represented with upper and lower limits, such as \( \int_{a}^{b} f(x) \, dx \), and are used to calculate areas under curves or the total accumulated value between two points.
Understanding these distinctions assists students in applying the correct integration approach, depending on whether they are interested in the general antiderivative or the specific value across a range. This foundation is crucial for comprehending the broad scope of calculus applications.
On the other hand, a definite integral defines the integration over a specific interval and provides a single numerical value rather than a function. Definite integrals are represented with upper and lower limits, such as \( \int_{a}^{b} f(x) \, dx \), and are used to calculate areas under curves or the total accumulated value between two points.
Understanding these distinctions assists students in applying the correct integration approach, depending on whether they are interested in the general antiderivative or the specific value across a range. This foundation is crucial for comprehending the broad scope of calculus applications.
