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Chapter 7: Problem 9

\(4-30\) Evaluate the integral. $$\int \frac{d x}{\sqrt{x^{2}+16}}$$

Short Answer

Expert verified
The integral \(\int \frac{d x}{\sqrt{x^{2}+16}}\) evaluates to \(\ln \left| \frac{x}{4} + \sqrt{1 + \frac{x^2}{16}} \right| + C\).

Step by step solution

01

Identify the Integral Type

The given integral \(\int \frac{d x}{\sqrt{x^{2}+16}}\) is of the form \(\int \frac{d x}{\sqrt{x^{2}+a^2}}\), which suggests a trigonometric substitution.
02

Choose a Trigonometric Substitution

To simplify the integral, use the substitution \(x = 4\tan \theta\). This leads to \(dx = 4\sec^2 \theta \, d\theta\), and helps simplify the expression under the square root.
03

Substitute and Simplify the Integral

Substitute \(x = 4\tan \theta\) into the integral:\[ \int \frac{4\sec^2 \theta \, d\theta}{\sqrt{16\tan^2 \theta + 16}} = \int \frac{4\sec^2 \theta \, d\theta}{\sqrt{16(\tan^2 \theta + 1)}} \]Since \(\tan^2 \theta + 1 = \sec^2 \theta\), the expression simplifies to:\[ \int \frac{4\sec^2 \theta \, d\theta}{4\sec \theta} = \int \sec \theta \, d\theta \]
04

Evaluate the Simplified Integral

The integral \(\int \sec \theta \, d\theta\) is a standard integral:\[ \int \sec \theta \, d\theta = \ln |\sec \theta + \tan \theta| + C \] where \(C\) is the integration constant.
05

Back-Substitute \(\theta\) in Terms of \(x\)

From \(x = 4\tan \theta\), we have \(\tan \theta = \frac{x}{4}\). In terms of \sec \theta, \(\sec \theta = \sqrt{1 + \tan^2 \theta} = \sqrt{1 + \frac{x^2}{16}}\).Therefore:\[ \ln |\sec \theta + \tan \theta| = \ln \left|\sqrt{1 + \frac{x^2}{16}} + \frac{x}{4}\right| \]
06

Write the Final Answer

Substitute back in the expression: The evaluated integral is:\[ \int \frac{d x}{\sqrt{x^{2}+16}} = \ln \left| \frac{x}{4} + \sqrt{1 + \frac{x^2}{16}} \right| + C \]

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Key Concepts

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

Definite Integral
A definite integral is an important concept in calculus used to compute the area under a curve between two points. Unlike indefinite integrals, which have a general form, definite integrals provide the exact accumulation of a quantity. It is often represented as:
  • \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of integration.
  • These limits define the interval over which we are calculating the area.
In the context of our exercise, although we are working with an indefinite integral, understanding definite integrals aids in understanding problems where specific bounds are given. They involve techniques like evaluating the antiderivative at these limits and subtracting the results, enhancing our understanding of integration's practical application.
Integration Techniques
Integration is a fundamental calculus tool for finding antiderivatives or areas under curves, and various techniques make this process easier. Each technique works best for different forms of integrals:
  • Trigonometric Substitution: As used in this exercise, substitute variables with trigonometric functions to simplify integrals with certain forms, such as \( \sqrt{x^2 + a^2} \).
  • Substitution: Replace expressions with a single variable to simplify an integral. This helps in making complicated integrals more manageable.
  • Integration by Parts: Useful for products of functions, using the formula \( \int u \, dv = uv - \int v \, du \).
Choosing the right method can change an impossible integral into a straightforward one. In our exercise, by substituting \( x = 4\tan \theta \), we transformed the original integral into one involving secants, a standard trigonometric form.
Antiderivative
An antiderivative is essentially the reverse process of differentiation, and it plays a key role in solving integrals:
  • Finding an antiderivative involves determining a function whose derivative matches the given function.
  • For example, if the derivative of a function \( F(x) \) is \( f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \).
With trigonometric substitution, we simplified \( \int \sec \theta \, d\theta \) to find \( \ln |\sec \theta + \tan \theta| + C \), which is an antiderivative of the integrand. In integration, the ultimate goal is often to identify or compute an antiderivative.
Integration Constant
When calculating an indefinite integral, which represents a family of functions, we include an integration constant, typically denoted as \(C\):
  • This constant accounts for any constant terms that could have been lost during differentiation.
  • Indefinite integration gives a general solution, representing all possible antiderivatives, which differ by this constant.
In our example, after finding the antiderivative, adding \( C \) gives the complete solution. It fills the gap left by the absence of boundary limits, showing the integral isn't restricted to a single function. This emphasizes the fundamental nature of integration as capturing all possible solutions to a differential equation.

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