Problem 25 - Thomas Calculus

Question

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{d s}{\left(9-s^{2}\right)^{2}}\)

Step-by-Step Solution

Answer

\(\int \frac{ds}{(9-s^2)^2} = \frac{s}{18(9-s^2)} + \frac{1}{54}\arcsin\left(\frac{s}{3}\right) + C\).

Step 1: Recognize the Form of the Integral

The integral we have is \(\int \frac{ds}{(9-s^{2})^{2}}\). This resembles a standard form usually found in a table of integrals: \(\int \frac{dx}{(a^2-x^2)^2}\) or a similar format. Our goal is to match this with the appropriate table entry.

Step 2: Identify Constants in Integral

In our integral, \(a\) is a constant, and we compare \(9\) and \(s^2\) with the general form. Here, we recognize that \(9 = 3^2\), so \(a = 3\). Our integral now matches the form \(\int \frac{ds}{(3^2-s^{2})^{2}}\).

Step 3: Find Matching Integral from Table

Referring to the table of integrals, we look for an entry that matches our form \(\int \frac{ds}{(a^2-s^2)^{2}}\). The table entry might suggest using a specific substitution or provide a direct formula.

Step 4: Apply the Formula from the Table

Assuming the table provides: \(\int \frac{dx}{(a^2-x^2)^2} = \frac{x}{2a^2(a^2-x^2)} + \frac{1}{2a^3}\arcsin\left(\frac{x}{a}\right) + C\). We apply this to our integral with \(a = 3\), so our result becomes: \(\frac{s}{2(3^2)(9-s^2)} + \frac{1}{2(3)^3}\arcsin\left(\frac{s}{3}\right) + C\).

Step 5: Simplify the Expression

Simplify the expression: \(\frac{s}{18(9-s^2)} + \frac{1}{54}\arcsin\left(\frac{s}{3}\right) + C\). This is often the final solution as provided in the table.

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function, which is a fundamental tool in calculus. The primary goal is to determine the antiderivative or the area under a curve.
Several techniques exist to tackle various integrals, including:

• Substitution: This involves replacing a part of the integral with a single variable to make the integration manageable. It's often used when the integral contains a function and its derivative.
• Integration by Parts: This is useful for products of functions, relying on the rule: \( \int u \, dv = uv - \int v \, du \. \).
• Partial Fraction Decomposition: Useful for rational functions, this technique splits a complex fraction into simpler parts.

For the exercise at hand, recognizing the form of the integral as one found in a table of integrals is a technique in itself. It avoids lengthy calculations by matching a known form and applying pre-determined solutions.

Table of Integrals

A table of integrals is a collection of formulas for integrals of common function types. These tables come in handy to quickly solve integrals that match known patterns.
To utilize a table, follow these steps:

• Identify the Form: Compare your integral with those in the table. Look for similar structures, terms, and forms.
• Match the Constants: Check if constants in the integral can be identified with those in a form from the table. Adjust your integral to fit this form if possible.
• Apply the Formula: Once a match is found, directly apply the corresponding formula from the table to solve the integral.

In the provided example, the integral \( \int \frac{ds}{(9-s^{2})^{2}} \) matches with a common form found in the table, enabling a straightforward solution by utilizing known formulas.

Trigonometric Substitution

Trigonometric substitution is a technique used in integration to handle integrals involving quadratic expressions, especially those of the form \( a^2 - x^2 \), \( a^2 + x^2 \), and \( x^2 - a^2 \).
The idea is to substitute a trigonometric function for the variable to simplify the integral. For example:

• For \( \sqrt{a^2 - x^2} \), use \( x = a \sin(\theta) \).
• For \( \sqrt{a^2 + x^2} \), use \( x = a \tan(\theta) \).
• For \( \sqrt{x^2 - a^2} \), use \( x = a \sec(\theta) \).

In the given exercise, though not necessarily applied directly, understanding this technique can provide insight into why certain integral forms exist in tables, where direct trigonometric identities or transformations might be used to derive solutions.