Problem 2 - Essential Calculus

Question

Use a table of integrals with forms involving $$\sqrt{u^{2} \pm a^{2}}$$ to find $$\int \frac{\sqrt{x^{2}-9}}{3 x} d x$$

Step-by-Step Solution

Answer

The solution to the integral \(\int \frac{\sqrt{x^{2}-9}}{3 x} dx\) is \(\sqrt{x^2 - 9} - 3 \ln |x + \sqrt{x^2 - 9}|\).

Step 1: Recognize the standard form

Recognize the given integral \(\int \frac{\sqrt{x^{2}-9}}{3 x} dx\) as the form \(\int \frac{\sqrt{u^{2}-a^{2}}}{u} du\). In this case, \(u = x\) and \(a = 3\).

Step 2: Apply the formula

From the table of integrals, we know that \(\int \frac{\sqrt{u^{2}-a^{2}}}{u} du = \sqrt{u^2 - a^2} - a \ln |u + \sqrt{u^2 - a^2}|\). Now, insert \(u = x\) and \(a = 3\) into the formula.

Step 3: Simplify the result

After substitution, the integral becomes \(\sqrt{x^2 - 3^2} - 3 \ln |x + \sqrt{x^2 - 3^2}|\). This can further be simplified as \(\sqrt{x^2 - 9} - 3\ln |x + \sqrt{x^2 - 9}|\).

Key Concepts

Definite Integrals

Definite integrals are used to calculate the area under a curve between two specific points on the x-axis. Unlike indefinite integrals, which include an arbitrary constant, definite integrals provide an exact number.

When you calculate a definite integral, it involves taking the antiderivative of a function and evaluating it at the upper and lower bounds of the interval. Here's a basic rundown of the process:

• Find the antiderivative of the given function.
• Evaluate the antiderivative at the upper limit of integration.
• Evaluate the antiderivative at the lower limit of integration.
• Subtract the lower value from the upper value to determine the area under the curve.

In problems involving definite integrals, often the area may represent something physical like distance, mass, or other quantities. Understanding definite integrals is crucial for interpreting an integral result in practical scenarios.

Indefinite Integrals

Indefinite integrals are all about finding a family of functions whose derivative is the given function. When you see an indefinite integral, it doesn't come with bounds, meaning the result will include a constant of integration, typically denoted as "C."

This is because derivatives get rid of constants during differentiation so we add "C" to our antiderivative to account for all possible original functions.

• The integral of a constant, say "a," is "ax + C."
• When integrating a function from a formula, use known rules or formulas, like power rule, where \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\).

Indefinite integrals help formulate general solutions to problems, showcasing how a function can transform over time or space. They lay the groundwork for finding definite integrals by giving us the antiderivative needed for further calculations.

Integral Tables

Integral tables are essential tools for finding the antiderivatives of complex functions. These tables list various common functions and their integrals, saving time and effort by providing instant access to difficult integral results.

Using an integral table allows you to quickly match your function to a known form. You can simply look up the integral of a known formula and apply it directly to your problem.

• Make sure the function pattern is exactly the same as in the table to use the corresponding integral.
• Substitute the relevant variables from your function into the variables/form given in the table.
• Modify constants accordingly to fit your specific problem.

In the given exercise, an integral table was used to solve \(\int \frac{\sqrt{x^{2}-9}}{3 x} dx\) by recognizing it as a specific form of \(\int \frac{\sqrt{u^{2}-a^{2}}}{u} du\). This demonstrates how tables simplify complex problems by providing direct solutions based on standard integral forms.