Problem 4 - Calculus of a Single Variable

Question

Use a table of integrals with forms involving \(\sqrt{a^{2}-u^{2}}\) to find the indefinite integral. \(\int \frac{\sqrt{64-x^{4}}}{x} d x\)

Step-by-Step Solution

Answer

The indefinite integral of \(\int \frac{\sqrt{64-x^{4}}}{x} dx\) is \(-32 \ln |\frac{x + \sqrt{64-x^{4}}}{x}| + C\).

Step 1: Identify the form of integral

Identify the given integral \(\int \frac{\sqrt{64-x^{4}}}{x} dx\) as being similar to the form \(\int \frac{\sqrt{a^{2}-u^{2}}}{u} du\). This is done by observing that \(64-x^{4}\) can be rewritten as \(8^{2}-x^{2^{2}}\), which resembles \(a^{2}-u^{2}\), and noting the similarity in their forms.

Step 2: Make the substitution

To transform the integral into the identified form, make a substitution: let \(u=x^{2}\), therefore \(du = 2x dx\) and \(x = \sqrt{u}\) . We can write our integral as \(\int \frac{\sqrt{64-u^{2}}}{\sqrt{u}} \frac{du}{2}\), which simplifies to \(\frac{1}{2}\int \frac{\sqrt{64-u^{2}}}{\sqrt{u}} du\). Now, the integral matches the identified form, where \(a = 8\) and \(u = x^{2}\).

Step 3: Apply the integral formula

Applying the standard integral formula for this form \(\int \frac{\sqrt{a^{2}-u^{2}}}{\sqrt{u}} du = -\frac{a^{2}}{2} \ln |\frac{\sqrt{u} + \sqrt{a^{2}- u^{2}}}{\sqrt{u}}|\), substitute \(a = 8\) and \(u = x^{2}\) to obtain \(-8^{2}/2 \ln |\frac{\sqrt{x^{2}} + \sqrt{8^{2}- x^{2^{2}}}}{\sqrt{x^{2}}}\).

Step 4: Simplify the expression

Simplify the expression to obtain \(-32 \ln |\frac{x + \sqrt{64-x^{4}}}{x}|\).

Step 5: Write down the indefinite integral

The indefinite integral of \(\int \frac{\sqrt{64-x^{4}}}{x} dx\) is \(-32 \ln |\frac{x + \sqrt{64-x^{4}}}{x}| + C\), where C represents the constant of integration.

Key Concepts

Integration Techniques

Integration is a crucial tool in calculus, used to find areas, volumes, and other quantities that accumulate. An important aspect of integration is choosing the right technique to solve different types of integrals. Some common integration techniques include:

• Substitution: Replacing a variable with another expression to simplify the integral.
• Partial Fractions: Decomposing complex fractions to simpler ones that are easier to integrate.
• Integration by Parts: A method based on the product rule for derivatives.
• Trigonometric Substitution: Using trigonometric identities to simplify integrals involving square roots.

Each technique requires careful analysis of the integral's form. In the given exercise, substitution is used as a powerful method to rewrite the integral into a recognizable form that fits a solution found in integral tables. By turning the problem into a standard expression, the solution becomes more approachable and simpler to compute.

Trigonometric Substitution

Trigonometric substitution is employed when an integral involves square roots of expressions like \(a^2-u^2\), \(a^2+u^2\) or \(u^2-a^2\). This technique leverages trigonometric identities to simplify the integral. Here's how it works:

• For \(a^2-u^2\), use \(u = a\sin\theta\), transforming the expression under the square root into \(a^2\cos^2\theta\).
• For \(a^2+u^2\), use \(u = a\tan\theta\), simplifying to \(a^2\sec^2\theta\).
• For \(u^2-a^2\), use \(u = a\sec\theta\), resulting in \(a^2\tan^2\theta\).

In the provided exercise, even though direct trigonometric substitution is not applied, recognizing forms that fit this technique aids in setting up the substitution that simplifies the integral. Understanding where and how to employ these trigonometric transformations opens up various ways to tackle integration problems.

Integral Tables

Integral tables are reference tools that list common integrals and their solutions. These tables are particularly useful when dealing with complex integrals or when quick solutions are needed. They provide formulas and solutions for:

• Basic integrals like \(\int x^n \, dx\).
• Trigonometric integrals including \(\int \sin x \, dx\) and \(\int \cos x \, dx\).
• Exponential integrals such as \(\int e^x \, dx\).
• Special forms like \(\int \frac{\sqrt{a^2-u^2}}{u} \, du\).

In the exercise, a table of integrals helps identify and solve an integral involving a square root form. By relating the problem to a known solution in the table, the process becomes much simpler. It's like having a mathematical dictionary that translates complex integrals into understandable terms. Knowing how to use these tables effectively is an invaluable skill for tackling indefinite integrals efficiently.