Problem 1 - Thomas Calculus

Question

Evaluate the integrals. $$\int \frac{d x}{\sqrt{9+x^{2}}}$$

Step-by-Step Solution

Answer

The integral evaluates to \( \ln\left|\sqrt{9+x^2} + x\right| + C \).

Step 1: Recognize the Integral Form

The integral in question is \( \int \frac{d x}{\sqrt{9+x^{2}}} \). This is a standard form of integral that can be solved using a trigonometric substitution. The form matches the integral \( \int \frac{d x}{\sqrt{a^2+x^2}} \), where \( a = 3 \).

Step 2: Choose Appropriate Trigonometric Substitution

For the integral \( \int \frac{d x}{\sqrt{9+x^{2}}} \), use the substitution \( x = 3 \tan(\theta) \). This substitution is effective because it uses the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \).

Step 3: Compute Differential in Substitution

After substituting \( x = 3 \tan(\theta) \), find the differential \( d x = 3 \sec^2(\theta) d\theta \). Substitute these into the integral.

Step 4: Apply the Substitution to the Integral

Replace \( x \) in the integral: \( \int \frac{3 \sec^2(\theta) d\theta}{\sqrt{9 + 9 \tan^2(\theta)}} \). Simplify the denominator using \( 9 + 9\tan^2(\theta) = 9\sec^2(\theta) \). Thus, the integral becomes \( \int \frac{3 \sec^2(\theta) d\theta}{3 \sec(\theta)} = \int \sec(\theta) d\theta \).

Step 5: Integrate the Trigonometric Integral

The resulting integral is \( \int \sec(\theta) d\theta \). The integral of \( \sec(\theta) \) is \( \ln|\sec(\theta) + \tan(\theta)| + C \), where \( C \) is the constant of integration.

Step 6: Substitute Back to Original Variable

Revert the substitution by recalling \( x = 3\tan(\theta) \), which gives \( \tan(\theta) = \frac{x}{3} \). Hence, \( \sec(\theta) = \sqrt{1 + \tan^2(\theta)} = \sqrt{1 + \left(\frac{x}{3}\right)^2} = \frac{\sqrt{9+x^2}}{3} \). Substitute these back into the integration result to obtain \( \ln\left|\frac{\sqrt{9+x^2}}{3} + \frac{x}{3}\right| + C \).

Step 7: Simplify the Result

The expression simplifies to \( \ln\left|\frac{\sqrt{9+x^2} + x}{3}\right| + C \). This can further be simplified to \( \ln\left|\sqrt{9+x^2} + x\right| - \ln(3) + C \) for clarity, but \( \ln(3) \) can be absorbed into the constant as \( C' \), leading to the final integral result: \( \ln\left|\sqrt{9+x^2} + x\right| + C' \).

Key Concepts

Integration Techniques

In calculus, integration techniques are diverse strategies used to compute integrals. A common technique is **Trigonometric Substitution**. This involves replacing variables in a given integral with trigonometric functions to simplify the expression.
For example, the integral \( \int \frac{d x}{\sqrt{9+x^{2}}} \) can be approached by recognizing it matches the form \( \int \frac{d x}{\sqrt{a^2+x^2}} \), allowing us to use a substitution like \( x = 3 \tan(\theta) \).
This substitution works because the trigonometric identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \) simplifies the integral's denominator. Transforming the function eases its evaluation, eventually leading us to an integral in terms of \( \theta \), such as \( \int \sec(\theta) \, d\theta \). Finally, reverting back to the original variable finishes the procedure. Understanding and distinguishing when to employ such techniques is essential in calculus as it unlocks more potential solutions.

Definite Integrals

Definite integrals, contrasting with indefinite integrals, provide the total accumulation of a function over a specific interval. In our exercise, discovering an indefinite integral involving trigonometric substitution was the focus, but such methods can extend to definite forms.
When you evaluate a definite integral from \( a \) to \( b \), you're implementing two major steps: computing the antiderivative and applying the **Fundamental Theorem of Calculus**. This theorem allows us to evaluate the integrals at the specific limits \( a \) and \( b \), culminating in a precise numeric result representing area.
Techniques like substitution, including trigonometric substitution, provide tools categorized as

• Substitution or change of variables,
• Integration by parts,
• Partial fractions, and more.

Each approach, crucial to mastery, can help tackle complex integral problems, particularly those with specific bounds.

Calculus Problem Solving

Problem-solving in calculus often involves a blend of various concepts and techniques to arrive at a solution. Understanding when and how to apply these methods can be enhanced by recognizing patterns in the functions involved.
In the given problem, the use of **Trigonometric Substitution** defines a clearer path to solving the integral. Identifying the integral model allowed the substitution \( x = 3 \tan(\theta) \), making it feasible to simplify and solve it effectively.
Key components of solving calculus problems include:

• Recognizing standard forms of integrals,
• Choosing suitable techniques like substitution,
• Simplifying complex expressions,
• Substituting back to original variables correctly.

By practicing these methods, students can enhance their problem-solving skills. Each solution builds a deeper understanding of calculus concepts, promoting confidence in tackling diverse and more complex mathematical challenges.