Problem 2 - Thomas Calculus

Question

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{d x}{x \sqrt{x+4}}\)

Step-by-Step Solution

Answer

The integral evaluates to \( \ln|\sqrt{x} + \sqrt{x+4}| + C \).

Step 1: Identify a Suitable Integral

Refer to the table of integrals in the textbook to find a suitable entry that resembles the integral \( \int \frac{dx}{x \sqrt{x+4}} \). Typically, such tables have integrals in the form \( \int \frac{dx}{x \sqrt{ax+b}} \) with a specific solution format.

Step 2: Match Parameters

Compare the given integral \( \int \frac{dx}{x \sqrt{x+4}} \) with the table form. Here, we have \( a = 1 \) and \( b = 4 \). The general form solution might look like \( \frac{1}{\sqrt{a}} \ln |\sqrt{a}x+\sqrt{ax+b}+C| \).

Step 3: Substitute and Evaluate

Substitute \( a = 1 \) and \( b = 4 \) into the solution from the table of integrals. The solution to our integral is \( \ln|\sqrt{x}+\sqrt{x+4}|+C \), where \( C \) is the constant of integration.

Key Concepts

Table of Integrals

When performing integration, especially in calculus, having a quick reference like a table of integrals can save you time and effort. These tables often present integrals in standard forms, along with their evaluated results. For students, this is a valuable tool to quickly find an antiderivative without having to work through complex calculations every time.
However, to effectively use a table of integrals:

• Identify the structure of the given integral.
• Locate the matching standard form in the table.
• Apply the corresponding solution while keeping track of any parameters or constants specific to your problem.

In the context of the exercise above, that means recognizing that \(\int \frac{dx}{x \sqrt{x+4}}\) matches a form found in these tables. This involves substituting known parameters, which in this case are the values of \(a\) and \(b\). This allows you to plug into the predetermined formula from the table, thus simplifying the entire process.

Integration Techniques

Integration techniques help in solving integrals that may not be straightforward by naturally decomposing, transforming, or considering substitutions. Each technique is essential in tackling different types of functions and their compositions. Common techniques include:

• Substitution: Changing variables to simplify the integral.
• Integration by Parts: Useful when the integrand is a product of functions.
• Partial Fraction Decomposition: For rational functions in a complex form.

The step-by-step solution utilizes one powerful technique—matching with a known standard integral form. Finding such forms is a critical skill when integrals resist simple algebraic manipulation. Here, it involved aligning the expression with \( \int \frac{dx}{x \sqrt{ax + b}} \) from the table, illustrating how recognizing patterns and forms is crucial to effective calculus problem-solving.

Constant of Integration

When you integrate a function, you often find an antiderivative that includes a constant of integration, denoted by \(C\). It represents the family of functions that differ by a constant, reflecting the indefinite nature of the integration process.
For example, integrating the function \(\frac{1}{x}\) results in \({\ln |x| + C}\). This constant is essential because it signifies that there are infinitely many antiderivatives, each shifted by some amount vertically.

• The constant ensures that every possible solution to the differential equation is captured.
• Without \(C\), important solutions might be overlooked, leading to potential issues in application.

In the given exercise, after evaluating the integral, the constant \(C\) is crucial as it completes the representation of the antiderivative \(\ln|\sqrt{x} + \sqrt{x+4}| + C\), thereby providing the full solution set for the integral.