Problem 16 - Applied Calculus

Question

Find each integral by using the integral table on the inside back cover. $$ \int \frac{1}{\sqrt{x^{2}-1}} d x $$

Step-by-Step Solution

Answer

\( \ln |x + \sqrt{x^2 - 1}| + C \)

Step 1: Identify the Integral Form

Examine the integral \( \int \frac{1}{\sqrt{x^{2}-1}} \, dx \). Identify its form by comparing it with standard forms found in an integral table.

Step 2: Match with Known Integral

Match the given integral with the standard integral form \( \int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \ln |x + \sqrt{x^2 - a^2}| + C \). Here, \(a = 1\).

Step 3: Apply the Integral Formula

Apply the formula from the table by substituting \(a=1\). The integral becomes \( \ln |x + \sqrt{x^2 - 1}| + C \).

Step 4: Write Down the Solution

The solution to the integral is \( \int \frac{1}{\sqrt{x^{2}-1}} \, dx = \ln |x + \sqrt{x^2 - 1}| + C \).

Key Concepts

Definite Integrals

Definite integrals give us a numerical value and are used to compute the area under a curve from one point to another. When dealing with definite integrals, we focus on calculating the net area between a function and the x-axis within a specific interval. The core idea is to evaluate the integral of a function from a lower limit to an upper limit.

Here are several important properties and concepts related to definite integrals that you should know:

• The notation for a definite integral is \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of integration.
• Evaluating a definite integral involves finding the antiderivative, denoted as \(F(x)\), of \(f(x)\) and computing \(F(b) - F(a)\).
• Definite integrals can have positive, negative, or zero values, depending on the area they represent above or below the x-axis.
• The Fundamental Theorem of Calculus links definite integrals with derivatives and provides a formula for evaluating them.

Mastering definite integrals is key to understanding how to accurately measure areas and is widely applied in practical sciences like physics and engineering.

Indefinite Integrals

Indefinite integrals are often a primary topic when diving into integral calculus. Unlike definite integrals which provide a numeric value, indefinite integrals express a family of functions. When integrating, we reverse the process of differentiation, leading to a new function plus a constant of integration, commonly represented as \(C\).

Several key elements differentiate indefinite integrals:

• The notation for an indefinite integral is \(\int f(x) \, dx\).
• Indefinite integrals do not have limits, which means the result is a general function plus \(C\).
• The constant \(C\) represents all possible constants that could have been derived when finding the antiderivative; it accounts for the family of solutions.
• To solve an indefinite integral, one must find an antiderivative of the function.

Indefinite integrals have numerous applications, especially in solving differential equations and finding original functions given their rates of change.

Integral Tables

Integral tables are an extremely handy tool for solving integrals quickly, especially when dealing with standard forms that appear often in calculus problems. These tables provide the antiderivatives of a wide variety of functions, and they help streamline the process of integration.

Here's what you need to understand about integral tables:

• Integral tables list common integrals and their corresponding antiderivatives.
• They are particularly useful for complex integrals where deriving the antiderivative manually would be too time-consuming.
• To use an integral table effectively, you must identify the form of the integral you are dealing with and find its match in the table.
• While integral tables simplify many problems, they may not include every possible function, thus requiring additional techniques for some cases.

By understanding how to use integral tables effectively, students can tackle calculus problems more efficiently, saving time and reducing potential errors in finding antiderivatives.