Problem 27 - Thomas Calculus

Question

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{\sqrt{4 x+9}}{x^{2}} d x\)

Step-by-Step Solution

Answer

Use substitution and identify forms similar to table entries. Result: an integrated expression in terms of \( x \).

Step 1: Identify the Integral

We need to evaluate the integral \( \int \frac{\sqrt{4x+9}}{x^2} \, dx \). The goal is to find a suitable substitution or recognize it as a form listed in a standard integral table.

Step 2: Check Table of Integrals for Standard Form

Look for an integral in the table that resembles \( \int \frac{\sqrt{ax+b}}{x^n} \, dx \). While there might not be an exact match, tables often have entries for integrals involving square roots and monomials.

Step 3: Apply a Suitable Substitution

Based on similar forms, one might try the substitution \( u = \sqrt{4x+9} \) (even if not in table, hypothesize), leading to \( du = \frac{2}{\sqrt{4x+9}} \, dx \) and solve for \( dx \). Express \( x \) in terms of \( u \) and \( dx \).

Step 4: Rewrite the Integral

Substitute back into the integral: replace \( \sqrt{4x+9} = u \) and \( dx \) with expressions involving \( u \). Adjust the integral accordingly, considering \( dx = \frac{u}{2} \, du \) and any remaining expressions.

Step 5: Perform the Integration

Integrate the simplified expression in terms of \( u \). This might require recognizing a simpler integration result from the table, such as integration rules for power functions or direct entries.

Step 6: Back-Substitute and Simplify

After integrating in terms of \( u \), substitute back the expression for \( u = \sqrt{4x+9} \) to express the integral in terms of \( x \). Simplify the final result.

Step 7: Verify with an Example

Check the computed integral by differentiating it to ensure it matches the original integrand. This step confirms the integration process was correctly executed.

Key Concepts

Substitution Method

The substitution method is a powerful technique used to simplify the integration process. When an integral seems complex or doesn't directly match with standard forms, substitution helps by introducing a new variable, often denoted as \( u \). This allows for the transformation of the integral into a more manageable form.

Here's how it works:

• Identify a part of the integral that could be substituted. This is often the expression inside a root, or anything causing the complexity.
• Choose an expression to substitute: For example, let \( u = \, \text{some function of x} \). Calculate the derivative of this expression to find \( du \), which represents the differential element.
• Rewrite the integral completely in terms of \( u \), so every instance of \( x \) has a corresponding \( u \) expression.
• Integrate with respect to \( u \), and after finding the antiderivative, swap back to the original variable \( x \) using the initial substitution.

By following these steps, the complexity is often reduced, making the integral more straightforward to solve.

Standard Integrals

Standard integrals serve as a toolkit for solving a wide variety of integral problems. They provide pre-calculated solutions for different types of integrals that appear frequently. Imagine them as a shortcut that can save time and efforts when dealing with complicated calculations.

Key points include:

• Standard integrals cover basic functions such as powers, exponentials, logarithms, and trigonometric functions.
• They're typically found in the back of math textbooks or reference materials. However, memorizing some of the common forms can be a huge advantage.
• When faced with an integral, always scan standard forms to check if there's a direct match. This could speed up the process immensely as you can immediately apply the known outcome.
• Even if there's no direct match, knowing the standard integrals could assist in recognizing possible simplifications or conversions to align an integral closer to a standard form.

Using these integrals effectively requires practice but can drastically improve efficiency in solving calculus problems.

Table of Integrals

Tables of integrals are extensive lists found typically at the end of a calculus textbook. They include numerous integral formulas derived from various functions. These tables are essential when a direct integration isn't obvious and a step-by-step solution might involve complex substitutions.

Here's how to utilize them:

• Scan the table when you are stuck with an integral that looks different or challenging. Look for patterns or similar structures within the table entries.
• If no example perfectly matches, look for partial matches and adjust your integral using algebraic manipulation or substitution method to align it more closely.
• They are especially helpful for forms involving roots, exponents, trigonometric functions, and other non-polynomial behaviors you might come across.
• Tables of integrals can also serve as a confirmation tool to check your manually-derived solutions against known results.

With practice, using a table of integrals can become second nature, providing significant insights and shortcuts while tackling integrals.