Problem 33 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{1}{\sqrt{9+x^{2}}} d x\)

Step-by-Step Solution

Answer

The evaluated integral is \(\int \frac{1}{\sqrt{9+x^{2}}} d x = \ln |\frac{2\sqrt{9 + x^{2}} - 3}{\sqrt{9 + x^{2}}}| + C \)

Step 1: Identify the appropriate substitution

Since the integral contains a sum of squares in the radical, a standard method is to use a substitution using tangent. Let \( x = 3 \tan(\theta) \), such that \( d x = 3 \sec^{2}(\theta) d \theta \). The angle \(\theta\) satisfies \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \)

Step 2: Substitute and simplify

Substitute the expression of \( x \) into the integral to get: \[ \int \frac{1}{\sqrt{9(1+\tan^{2}(\theta))}} \cdot 3 \sec^{2}(\theta) d \theta = \int \sec(\theta) d\theta \]. Now, the integral simplifies to a well-known integral, the integral of the secant function.

Step 3: Evaluate the integral

The integral of the secant function \(\sec(\theta)\) is \( \ln | \sec(\theta) + \tan(\theta) | + C \). So, the integral becomes: \[ \int \sec(\theta) d\theta = \ln | \sec(\theta) + \tan(\theta) | + C \]

Step 4: Convert back to the original variable

But the answer needs to be in terms of \( x \), not \( \theta \). So, using the initial substitution, and recalling that \(\sec(\theta) = \frac{1}{\cos(\theta)}\) and \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), we can rewrite \(\sec(\theta) + \tan(\theta)\) as \[ \frac{1 + \sin(\theta)}{\cos(\theta)} = \frac{2}{2\cos(\theta)} - \frac{1 - \sin(\theta)}{2\cos(\theta)} = \frac{2}{1 + \cos(\theta)} - 1 = 2 \sec^{2}(\frac{\theta}{2}) - 1\].Given the substitution \(x = 3\tan(\theta)\), we know that \(\tan(\theta) = \frac{x}{3}\).So, \(\cos(\theta) = \frac{1}{\sqrt{1 + (\frac{x}{3})^{2}}}\) and \(\sec(\frac{\theta}{2}) = \sqrt{\frac{1 + \cos(\theta)}{2}} = \frac{2}{\sqrt{9 + x^{2}}}\).Substituting this back into the integral, it becomes:\[ \ln | \sec(\theta) + \tan(\theta) | + C = \ln | 2\cdot \frac{2}{\sqrt{9 + x^{2}}} - 1| + C = \ln |\frac{2\sqrt{9 + x^{2}} - 3}{\sqrt{9 + x^{2}}}| + C \].Hence, the final answer to the integral is \[ \int \frac{1}{\sqrt{9+x^{2}}} d x = \ln |\frac{2\sqrt{9 + x^{2}} - 3}{\sqrt{9 + x^{2}}}| + C \].

Key Concepts

Integration Techniques

When tackling integrals in calculus, the skill to identify the right technique is crucial. One common approach is trigonometric substitution, which is ideal for integrals involving square roots of sums or differences of squares. It works by transforming the original integral into a trigonometric one that is easier to evaluate.

Other integration techniques include integration by parts, partial fractions, and u-substitution. Each method has its suitability, depending on the function presented. For instance, integration by parts effectively handles products of functions, whereas partial fractions decompose complex rational functions.

Choosing the Right Method

Deciding on which technique to apply often requires practice and familiarity with the form of the functions involved. In the given exercise, trigonometric substitution was the method of choice due to the sqrt(a² + x²) structure.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to work backwards from the trigonometric ratios to the angles. Commonly used inverse trigonometric functions in calculus include arcsin, arccos, and arctan.

They are especially handy in integration when dealing with expressions that result from trigonometric substitutions, as they can help revert the transformed trigonometric expression back into terms of the original variable.

Why Use Inverse Trigonometry

In our integral solution above, we didn't utilize inverse trigonometric functions directly, but the concept is still involved implicitly as we transition back from the angle θ to the variable x. This reverse journey is where understanding and applying the principles of inverse trigonometry becomes essential.

Secant Function

In trigonometry, the secant (sec) is the reciprocal of the cosine function. Unlike the more commonly known sine and cosine functions, the secant function takes the hypotenuse over the adjacent side in a right triangle.

The integral of the secant function, which appears in the solution, has its own unique identity: ∫ sec(θ) dθ = ln | sec(θ) + tan(θ) | + C. Recognizing this identity is key to solving integrals that simplify to the form of a secant function.

Significance in Integration

Such identities are part of a broader toolkit that a student needs to memorize or understand to tackle integrals quickly and correctly. In our example, recognizing the secant integral saved steps and simplified the process.

Integral Calculus

Integral calculus is fundamentally about finding the function's antiderivative, which represents the accumulation of quantities like area, volume, and other concepts. The process of integration, which involves a variety of techniques including trigonometric substitution, is pivotal for solving problems in physics, engineering, and beyond.

As seen in this exercise, after using trigonometric substitutions, we integrate to find a function in terms of θ and then translate that back into the original variable x to complete the solution.

From Concept to Application

This example affirms how integral calculus does more than solve abstract mathematical problems—it also teaches us to transform and interpret expressions in multiple ways to reveal deeper insights, thereby playing a crucial role in the application of mathematics to real-world problems.