Question

In Problems 1-16, perform the indicated integrations. \(\int \frac{\sqrt{4-x^{2}}}{x} d x\)

Short answer

The integral evaluates to \( \sqrt{\frac{4-x^2}{x^2}} + C \).

Step-by-step solution

Identify Integration Method

The integrand involves a square root of a quadratic expression in terms of another variable. This suggests the use of trigonometric substitution. Specifically, for expressions like \( \sqrt{a^{2} - x^{2}} \), we can use substitution with \( x = a \sin(\theta) \).

Apply Trigonometric Substitution

We substitute \( x = 2\sin(\theta) \), which implies \( dx = 2\cos(\theta)d\theta \). Substituting into the integral gives.

Simplify the Integral

Replace \( \sqrt{4-x^2} \) with \( \sqrt{4 - (2\sin(\theta))^2} = \sqrt{4(1 - \sin^2(\theta))} = 2\cos(\theta) \). The integral becomes \( \int \frac{2\cos(\theta)}{2\sin(\theta)} \cdot 2 \cos(\theta) d\theta = \int \frac{2\cos^2(\theta)}{\sin(\theta)} d\theta \).

Simplify Further Using Trigonometric Identities

Simplify \( \int \frac{2\cos^2(\theta)}{\sin(\theta)} d\theta = \int 2 \cot(\theta) \cos(\theta) d\theta \). Use the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \).

Use Trigonometric Identity in Integral

Since \( u = \sin(\theta) \), \( du = \cos(\theta) d\theta \), rewrite the integral as \( \int \frac{2\cos^2(\theta)}{\sin(\theta)} d\theta = \int 2 \cos(\theta) \cot(\theta) d\theta = \int 2 \frac{\cos^2(\theta)}{\sin(\theta)} d\theta = \int 2u du \).

Perform the Integration

Integrate \( \int 2u du \) which equals \( u^2 \). Substitute back to get \( \cot(\theta)^2 + C \).

Back Substitution

We originally substituted \( x = 2\sin(\theta) \) so \( \theta = \arcsin\left(\frac{x}{2}\right) \) and \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \sqrt{\frac{4-x^2}{x^2}} \). Substitute back to get the answer.

Final Simplification

The integral evaluates to \( \sqrt{\frac{4-x^2}{x^2}} + C \) or simplified further using algebraic manipulation.

Key concepts

Integration Techniques

Integration often requires choosing the right technique, especially when faced with complex expressions. One such advanced technique involves trigonometric substitution, which is ideal for integrals involving square roots of quadratic expressions. The goal is to simplify these expressions by substituting a trigonometric function.
For instance, if you encounter an integrand resembling \( \sqrt{a^2 - x^2} \), the substitution \( x = a \sin(\theta) \) is typically effective. This substitution leverages the Pythagorean identity, converting the integral into a trigonometric form. Such conversions can greatly simplify the problem, making it easier to solve using standard trigonometric integrals.
It's always crucial to properly convert both the function and differential \( dx \), which becomes \( ad\cos(\theta) d\theta \). The main advantage of this technique is that it handles the square root efficiently by directly relating it to cosine, creating a tractable quadratic expression. This showcases how integration techniques can transform seemingly intricate calculus tasks into manageable computations.

Trigonometric Identities

Trigonometric identities play a pivotal role in integration techniques, especially when dealing with expressions that involve angles and trigonometric functions.
In the given exercise, the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) simplifies the expression \( \sqrt{4 - x^2} \) to a form involving the cosine function. Each identity serves a purpose. For instance:

• \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \): Useful for reducing expressions involving \( \sin^2 \).
• \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \): Helps simplify integrals by restating them in terms of \( \cos \) alone.

Utilizing these identities effectively can drastically reduce the complexity of the integration problem. Identifying the correct identity and applying it can aid in breaking down the function into integral components that are straightforward to manage, thus simplifying the computation. Recognizing and manipulating these identities correctly is a powerful skill in calculus.

Calculus Problem Solving

Solving calculus problems often requires a strategic approach, combining different techniques and knowledge areas. Trigonometric substitution is just one part of the toolkit, especially useful for simplifying integrals involving square roots and promoting an easier solution method.
The step-by-step nature of calculus problem-solving is crucial. Here is how you can systematically tackle such problems:

• Determine the appropriate method (e.g., trigonometric substitution for square roots).
• Simplify the integrand using identities.
• Convert back to the original variables once integration is done.

Throughout the problem-solving process, back-substitution ensures that solutions are expressed in terms of the original variables, providing answers that make physical sense, consistent with the problem's context. Ultimately, calculus problem-solving is about breaking down complex tasks into understandable parts, applying appropriate mathematical rules and identities, streamlining the process through effective substitutions, and assembling the solution piece by piece to arrive at the final answer.