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Chapter 8: Problem 35

Use integration tables to find the indefinite integral. \(\int \frac{x}{\sqrt{x^{4}-6 x^{2}+5}} d x\)

Short Answer

Expert verified
\( \frac{1}{2} Arcsec(x^{2}-3) + C \)

Step by step solution

01

Recognize Pattern

Recognize the integral as a derivative of the arcsecant function. The derivative of \( Arcsec(u) \) is \( \frac{1}{|u|\sqrt{u^{2}-1}} \). The given integral matches this pattern: \( \int \frac{u'}{|u|\sqrt{u^{2}-1}} du \), where \( u = x^{2} \), and \( x = \sqrt{u} \). Therefore, we will use these substitutions in the next steps.
02

Substitute variables

Rewrite the integral with the new variable \( u \) and its derivative \( du \). Here, by Chain Rule, the derivative \( du \) is \( 2xdx \). The integral thus becomes: \( \frac{1}{2} \int \frac{du}{\sqrt{u^{2}-6u+5}} \). Notice that the factor of 1/2 is factored out by the constant coefficient rule for integrals.
03

Complete the Square to match the pattern

Rewrite the expression \( u^{2}-6u+5 \) in the denominator as a perfect square. This is done by completing the square: \( u^{2}-6u+5 = (u-3)^{2}-4 \). Replace this expression in the integral. It now looks like this: \( \frac{1}{2} \int \frac{du}{\sqrt{(u-3)^{2}-4}} \). This pattern matches the derivative of \( Arcsec(u) \) where \( u \) is replaced by \( u-3 \).
04

Integration using Integral Tables

Recall that the integral of \( \frac{du}{|u|\sqrt{u^{2}-1}} \) is \( Arcsec(u) + C \). Here, \( u-3 \) is replaced by \( u \). Therefore, the integral is: \( \frac{1}{2} Arcsec(u-3) + C \).
05

Re-substitute \( u \) with \( x^{2} \)

Re-substitute \( u \) with \( x^{2} \) into the integral: \( \frac{1}{2} Arcsec(x^{2}-3) + C \).

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Key Concepts

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

Arcsecant Function
The arcsecant function, denoted as \( \text{Arcsec}(u) \), is the inverse of the secant function. This means it helps us find an angle whose secant is \( u \). In calculus, the derivative of the arcsecant function reveals unique patterns that are essential for integration. Specifically, if \( u \) is a differentiable function of \( x \), then the derivative of \( \text{Arcsec}(u) \) is given by \( \frac{1}{|u| \sqrt{u^2 - 1}} \).

  • The presence of \( |u| \) denotes absolute value, ensuring the function behaves correctly across its domain.
  • The \( \sqrt{u^2 - 1} \) indicates that the expression under square root must remain positive for real number operations.
Recognizing this pattern in integrals, as shown in the original solution, helps simplify complex expressions and integrate them efficiently.
Indefinite Integral
An indefinite integral represents a family of functions whose derivatives equal the integrand. It's written as \( \int f(x) \, dx = F(x) + C \), where \( C \) is the constant of integration. Indefinite integrals are essential because they provide a way to reverse differentiation.

  • The symbol \( \int \) signifies integration, aimed to 'accumulate' or 'summarize' the function \( f(x) \).
  • \( C \) emphasizes the general solution nature, capturing all possible outcomes from antiderivatives.
In the given exercise, finding the indefinite integral involves recognizing the function's relation to known patterns like that of arcsecant. This enables the application of integration techniques to solve the problem.
Substitution Method
The substitution method is a powerful technique in integration that simplifies integrals by transforming variables. It involves identifying a portion of the integral to substitute with a new variable. This technique changes the function into a form that's easier to integrate.

In the exercise:
  • The substitution \( u = x^2 \) is introduced. This choice stems from identifying \( x^2 \) as a central component of the integral's pattern.
  • The derivative \( du = 2x \, dx \) redefines the expression's variables, simplifying the integral as \( \frac{1}{2} \int \frac{du}{\sqrt{u^2 - 6u + 5}} \).
This step transforms the original problem into a more manageable form, paving the way for further techniques like completing the square.
Completing the Square
Completing the square is an algebraic method employed to rewrite quadratic expressions of the form \( ax^2 + bx + c \) into \( (x - h)^2 - k \) or similar forms. This transformation eases the process of integration and finding real roots or extrema.

  • In the provided solution, \( u^2 - 6u + 5 \) becomes \( (u - 3)^2 - 4 \).
  • This makes the expression inside the integral align with the known derivative form of the arcsecant function, \( \text{Arcsec}(u) \).
By using this technique, the problem becomes more navigable, leading smoothly into solving the integral by known methods.

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Most popular questions from this chapter

Proof Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b],\) then $$f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t$$

True or False? In Exercises \(85-88\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of \(f\) is symmetric with respect to the origin or the \(y\) -axis, then \(\int_{0}^{\infty} f(x) d x\) converges if and only if \(\int_{-\infty}^{\infty} f(x) d x\) converges.

Writing (a) The improper integrals $$ \int_{1}^{\infty} \frac{1}{x} d x \text { and } \int_{1}^{\infty} \frac{1}{x^{2}} d x $$ diverge and converge, respectively. Describe the essential differences between the integrands that cause one integral to converge and the other to diverge. (b) Sketch a graph of the function \(y=(\sin x) / x\) over the interval \((1, \infty) .\) Use your knowledge of the definite integral to make an inference as to whether the integral $$\int_{1}^{\infty} \frac{\sin x}{x} d x$$ converges. Give reasons for your answer. (c) Use one iteration of integration by parts on the integral in part (b) to determine its divergence or convergence.

Graphical Analysis In Exercises 85 and \(86, \operatorname{graph} f(x) / g(x)\) and \(f^{\prime}(x) / g^{\prime}(x)\) near \(x=0 .\) What do you notice about these ratios as \(x \rightarrow 0 ?\) How does this illustrate L'Hopital's Rule? $$ f(x)=e^{3 x}-1, \quad g(x)=x $$

Convergence or Divergence In Exercises \(53-62,\) use the results of Exercises \(49-52\) to determine whether the improper integral converges or diverges. $$ \int_{1}^{\infty} \frac{1}{x^{5}} d x $$
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