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Chapter 8: Problem 21

Finding an Indefinite Integral In Exercises \(19-40\) , use a table of integrals to find the indefinite integral. $$\int \frac{2}{x^{3} \sqrt{x^{4}-1}} d x$$

Short Answer

Expert verified
The indefinite integral of \( \int \frac{2}{x^{3} \sqrt{x^{4}-1}} \, dx \) is \( \text{arccosh}(x^{2}) + C \)

Step by step solution

01

Recognize the Form of the Integral

In this task, the integral is given by \( \int \frac{2}{x^{3} \sqrt{x^{4}-1}} \, dx \). Analyzing this, one can note that possibly a substitution method involving a trigonometric function might simplify it. Identifying substitution can crucially make the calculation process simpler.
02

Substitution

Choose a new variable \( u \), as per the form \( u = x^{2} \). This gives us \( du = 2x \, dx \) and \( dx = \frac{du}{2x} \), which can be simplifed as \( dx = \frac{du}{2u^{\frac{1}{2}}} \), considering \( x = u^{\frac{1}{2}} \). Now the original integral can be re-written with the new variable as: \( \int \frac{2}{u^{ \frac{3}{2}} \sqrt{u^{2} - 1}} \, \frac{du}{2u^{\frac{1}{2}}} \) which simplifies to \( \int \frac{1}{u \sqrt{u^{2} - 1}} \, du \).
03

Simplifying the Integral

In this form, it can now be seen that integral is in the form of \( \int \frac{1}{u \sqrt{u^{2} - 1}} \, du \), which is known from integral table as the result of arccosh(u). Thus, the solution of the integral is simply \( \text{arccosh}(u) + C\), where \(C\) is the constant of integration.
04

Back Substitution

Finally, we substitute \( u \) by the original variable \( x^{2} \), which results in the indefinite integral: \( \text{arccosh}(x^{2}) + C \)

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Key Concepts

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

Integration by Substitution
Integration by substitution is a powerful technique used to simplify complex integrals, making them easier to solve. It's like solving a puzzle by first turning it into a much simpler one. In this method, we choose a new variable, often denoted by \( u \), which helps to transform the integral into a simpler form.
  • The key idea is to replace a part of the integral by the new variable \( u \), so the integral becomes easier to evaluate. This involves identifying a part of the integral that can be substituted, simplifying the function.
  • For example, in the original equation \( \int \frac{2}{x^{3} \sqrt{x^{4}-1}} \, dx \), we cleverly pick \( u = x^{2} \), as it rearranges the complex expression into something manageable.
  • Next, we compute \( du \) based on our choice for \( u \), and then express \( dx \) in terms of \( du \). Substituting these into the original integral can turn a daunting task into a straightforward one.
Using integration by substitution is often the first step in simplifying a challenging integral. It opens the door to using further techniques if the integral still isn't simple enough to solve outright.
Trigonometric Substitution
Trigonometric substitution is a special case of substitution, particularly useful when dealing with expressions involving square roots. This involves substituting trigonometric identities into the integral to render it into a more standard form.
  • This technique is particularly used when integrals include expressions like \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \).
  • By transforming these into trigonometric identities, the difficulty of the integral is often greatly reduced. Trigonometric identities can simplify the expression under the square root.
In our example, identifying substrates like \( \sqrt{x^4 - 1} \), we may consider transformations that utilize trigonometric identities, giving easier-to-integrate expressions. Consider how replacing \( x \) with a trig function might transform the integral; it de-emphasizes the complications of square roots.
Integral Tables
Integral tables are a valuable resource in calculus. They contain a list of standard integrals that can help students solve complex integrals quickly.
  • The tables provide the antiderivatives of many common functions. Essentially, it's like having a cheat sheet where you match parts of your integral to one listed in the table.
  • They can save time when the integral is complex, or when an indefinite integral isn't easily solved through standard methods.
  • Instead of deriving every integral from scratch, you can look up similar patterns, easing the cumbersome process of integration.
In the provided exercise, after substitution, recognizing the resultant integral form allows us to directly apply a known result from integral tables. Specifically, it helps us conclude that \( \int \frac{1}{u \sqrt{u^2 - 1}} \, du \) matches the format of an integral that evaluates to \( \text{arccosh}(u) \). This application of integral tables significantly cuts down on the work, offering a direct route to the solution.

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Most popular questions from this chapter

In Exercises 49 and \(50,\) determine all values of \(p\) for which the improper integral converges. $$\int_{0}^{1} \frac{1}{x^{p}} d x$$

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