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Chapter 8: Problem 18

Using Formulas In Exercises \(17-20,\) use the Special Integration Formulas (Theorem 8.2 ) to find the indefinite integral. $$ \int \sqrt{4+x^{2}} d x $$

Short Answer

Expert verified
The indefinite integral of \(\sqrt{4+x^{2}}\) is \(\sqrt{4 + x^{2}} + 2 ln|x + \sqrt{4 + x^{2}}| + C\).

Step by step solution

01

Identify the Pattern

The given integral, \(\int \sqrt{4+x^{2}} dx \), follows the pattern of \(\int \sqrt{a^{2}+x^{2}} dx\), with \(a=2\).
02

Apply the Formula

Apply the theorem 8.2, which states \(\int \sqrt{a^{2}+x^{2}} dx = x/2 \sqrt{a^{2} + x^{2}} + a^{2}/2 ln|x + \sqrt{a^{2} + x^{2}}| + C\). Thus, the integral becomes \(2/2 \sqrt{4 + x^{2}} + 4/2 ln|x + \sqrt{4 + x^{2}}| + C\).
03

Simplify the Expression

Simplify the expression to yield \(\sqrt{4 + x^{2}} + 2 ln|x + \sqrt{4 + x^{2}}| + C\).

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Key Concepts

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

Special Integration Formulas
Special integration formulas are essential tools that simplify the process of finding indefinite integrals. These formulas help you quickly identify and solve integrals by recognizing common patterns. For example, integrating basic functions such as powers of x, exponential functions, and trigonometric functions can be made much simpler when using specific formulas.
  • These formulas reduce the complexity of integration.
  • They are particularly useful for more complicated functions.
  • Understanding these can greatly speed up calculations.
Grasping how these formulas work is key to tackling a broad range of integration problems efficiently.
Integration Patterns
Recognizing integration patterns is crucial for determining which special formula to apply. In the original exercise, the integral \(\int \sqrt{4+x^{2}}\, dx\) follows the pattern \(\int \sqrt{a^{2}+x^{2}}\, dx\). By identifying that \(a = 2\), you can effectively match the pattern to the correct formula.
  • Patterns act as the bridge to applying the right formula.
  • They help isolate parts of the integral to simplify it.
  • Identifying patterns saves time and avoids errors.
Developing the ability to spot these patterns will help you across various integral problems.
Theorem 8.2
Theorem 8.2 specifically addresses the integration of expressions involving square roots of sums of squares. It is a valuable theorem because it gives a straightforward method for finding integrals of this type. The formula stated in Theorem 8.2 is:\[\int \sqrt{a^{2}+x^{2}}\, dx = \frac{x}{2} \sqrt{a^{2} + x^{2}} + \frac{a^{2}}{2} \ln|x + \sqrt{a^{2} + x^{2}}| + C\]
  • This formula directly applies to integrals involving \(\sqrt{a^{2}+x^{2}}\).
  • It breaks down the integral into simpler, calculable parts.
  • Utilizing Theorem 8.2 simplifies handling square root integrals efficiently.
Becoming familiar with Theorem 8.2 is advantageous for solving similar problems with ease.
Square Root Integration
Square root integration comes into play when dealing with expressions containing square roots. These integrals pose unique challenges because they often involve more complex manipulation compared to standard polynomial integrals. In the exercise, integrating \(\sqrt{4+x^{2}}\) necessitated recognizing it as a specific pattern related to a known formula.
  • Square roots add layers of complexity.
  • Special techniques and formulas are often required.
  • Recognizing related patterns simplifies the process.
Mastering square root integration equips you with the skills to tackle a wider array of integral problems.

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Most popular questions from this chapter

L'Hopital's Rule State L'Hopital's Rule.

U-Substitution In Exercises 109 and 110 , rewrite the improper integral as a proper integral using the given u-substitution. Then use the Trapezoidal Rule with \(n=5\) to approximate the integral. $$ \int_{0}^{1} \frac{\sin x}{\sqrt{x}} d x, \quad u=\sqrt{x} $$

Evaluating an Improper Integral In Exercises \(33-48\) determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{4}^{\infty} \frac{\sqrt{x^{2}-16}}{x^{2}} d x $$

Comparison Test for Improper Integrals In some cases, it is impossible to find the exact value of an improper integral, but is important to determine whether the integral converges or diverges. Suppose the functions \(f\) and \(g\) are continuous and \(0 \leq g(x) \leq f(x)\) on the interval \([a, \infty) .\) It can be shown that if \(\int a_{a}^{\infty} f(x) d x\) converges, then \(\int_{a}^{\infty} g(x) d x\) also converges, and if \(\int_{a}^{\infty} g(x) d x\) diverges, then \(\int_{a}^{\infty} f(x) d x\) also diverges. This is known as the Comparison Test for improper integrals. (a) Use the Comparison Test to determine whether \(\int_{1}^{\infty} e^{-x^{2}} d x\) converges or diverges. (Hint: Use the fact that \(e^{-x^{2}} \leq e^{-x}\) for \(x \geq 1 . )\) (b) Use the Comparison Test to determine whether \(\int_{1}^{\infty} \frac{1}{x^{5}+1} d x\) converges or diverges. (Hint: Use the fact that \(\frac{1}{x^{5}+1} \leq \frac{1}{x^{5}}\) for \(x \geq 1 . )\)

Mathematical Induction Use mathematical induction to verify that the following integral converges for any positive integer \(n .\) $$\int_{0}^{\infty} x^{n} e^{-x} d x$$
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