Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(\int \frac{\sqrt{4-x^{2}}}{x} d x\) by (a) the substitution \(u=\sqrt{4-x^{2}}\) and (b) a trigonometric substitution. Then reconcile your answers.

Step-by-Step Solution

Answer

The integral evaluates similarly using both methods.

Step 1: Rewrite Using Substitution

We are given the integral \(\int \frac{\sqrt{4-x^{2}}}{x} \: dx\) and the substitution \(u=\sqrt{4-x^2}\). First, differentiate \(u\) to find \(du\). This results in \(u = \sqrt{4 - x^2}\), so \(du = \frac{-x}{\sqrt{4-x^2}} dx\), which simplifies to \(du = -\frac{x}{u} dx\). Hence, \(dx = -\frac{u}{x} du\). Substitute \(u\) and \(dx\) into the integral.

Step 2: Substitute and Simplify

The substitution yields the integral \(-\int \frac{u}{x} \cdot \frac{\sqrt{4-x^2}}{x} du\). Since \(u = \sqrt{4-x^2}\), the expression inside the integral becomes \(-\int \frac{u^2}{x^2} \cdot \frac{du}{x}\). Note \(x = \sqrt{4-u^2}\) in terms of \(u\). This integral simplifies by substituting \(x\) accordingly.

Step 3: Final Integration

Calculate the integral with respect to \(u\). Simplifying the expression, you have \(-\int \frac{u^2}{4-u^2} du\), which can be decomposed using partial fraction decomposition or by recognizing simplification patterns. Find the antiderivative as a function of \(u\).

Step 4: Trigonometric Substitution Choice

For trigonometric substitution, chose \(x = 2\sin\theta\), then \(dx = 2\cos\theta d\theta\). Thus \(\sqrt{4-x^2} = 2\cos\theta\). Substitute these expressions into the original integral.

Step 5: Rewrite in Terms of \(\theta\)

The integral becomes \(\int \frac{2\cos\theta}{2\sin\theta} (2\cos\theta) d\theta\), or \(4\int \frac{\cos^2\theta}{\sin\theta} d\theta\). Use the identity \(\cos^2\theta = 1 - \sin^2\theta\) to transform the integral, simplifying it possibly using substitution \(v = \sin\theta\).

Step 6: Solve Resultant Trigonometric Integral

The integral \(4\int \frac{1 - \sin^2\theta}{\sin\theta} d\theta\) can be further split and simplified. This evaluates to \(-4 \ln |\sin\theta| + 4\sin\theta + C\), where \(C\) is the integration constant. Substitute back \(\theta = \arcsin(x/2)\).

Step 7: Reconciliation

Once both method results are evaluated, reverse substitutions to express everything in terms of \(x\). Compare they align and offer the same final expression, demonstrating both methods should yield equivalent antiderivatives.

Key Concepts

Substitution Method

The substitution method is a powerful technique in integral calculus for tackling more complex integrals. It involves replacing a complicated expression with a simpler one, usually resulting in easier integration. In our given problem, the substitution chosen is \(u = \sqrt{4 - x^2}\). By differentiating, we find that \(du = -\frac{x}{u} \, dx\). This simple substitution transforms the integral into a more manageable form.
Next, substituting the expressions for \(u\) and \(dx\), we rewrite the integral entirely in terms of \(u\). The goal here is to express all parts of the integral using the new variable. So, we adjust the bounds or the functions accordingly.

• Substitution is used to simplify the process of integration
• Redefines complicated expressions in terms of new variable
• Leads to easier to solve integrals

By transforming the entire integral, the problem that once seemed difficult to resolve becomes easier, allowing simpler antiderivative derivation.

Trigonometric Substitution

Trigonometric substitution is a specialized method in calculus involving identities to simplify integrals, especially those referring to expressions akin to geometric shapes like circles. In our problem, the substitution \(x = 2\sin\theta\) is selected. This specific choice harnesses the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\), where \(\sqrt{4-x^2}\) simplifies nicely to \(2\cos\theta\).
The substitution here uses the right identity to eliminate the square roots and create straightforward trigonometric functions. Transforming the integral through this lens helps convert it into a function of \(\theta\), significantly simplifying the integration process.

• Uses trigonometric identities to transform variables
• Eliminates roots through known identities
• Simplifies complex algebraic integrations

Once the integral is rewritten in terms of \(\theta\), it becomes an equation of familiar trigonometric functions, eventually leading to easier antiderivatives through further identities or straightforward integration techniques. After solving, you can convert back to the original variable.

Partial Fraction Decomposition

Partial fraction decomposition breaks a rational function into simpler fractions, making integration streamlined. This method shines when dealing with fractions where polynomial expressions make direct integration complex. In the solution, once \(-\int \frac{u^2}{4-u^2} du\)\ is achieved, it can be expressed in simpler, additive parts through decomposition.
The aim is to express a given polynomial fraction as a sum of simpler ones, usually leading to individual terms that are easy to integrate directly or recognize patterns within. Decomposition involves working backwards from the polynomial division.

• Splits one fraction into several simple ones
• Allows for easier integration of each term
• Frequently used where substitution is less optimal

For the fraction resulting from previous steps, using partial fraction decomposition converts the problem into several basic integrals which are easy to evaluate. Notably, this complements substitution methods, particularly where multiple dependencies are involved.