Problem 4 - Calculus - AP Edition

Question

Evaluate the integral. $$\int \frac{d x}{x^{2} \sqrt{9-x^{2}}}$$

Step-by-Step Solution

Answer

\(-\frac{1}{9} \cdot \frac{\sqrt{9 - x^2}}{x} + C\)

Step 1: Choose Trigonometric Substitution

To evaluate the integral \( \int \frac{dx}{x^2 \sqrt{9-x^2}} \), observe that the expression under the square root is \( 9 - x^2 \). This is similar to \( 9 \cos^2(\theta) \) in trigonometric substitution. Let \( x = 3 \sin(\theta) \). This implies \( dx = 3 \cos(\theta) \, d\theta \) and \( \sqrt{9 - x^2} = 3 \cos(\theta) \).

Step 2: Substitute and Simplify

Substitute \( x = 3 \sin(\theta) \) and \( dx = 3 \cos(\theta) \, d\theta \) into the integral to get\[\int \frac{3 \cos(\theta) \, d\theta}{(3 \sin(\theta))^2 \cdot 3 \cos(\theta)} = \int \frac{3 \cos(\theta) \, d\theta}{9 \sin^2(\theta) \cdot 3 \cos(\theta)}.\]Cancel the \( 3 \cos(\theta) \) terms to simplify to\[\int \frac{d\theta}{9 \sin^2(\theta)} = \frac{1}{9} \int \csc^2(\theta) \, d\theta.\]

Step 3: Integrate Using Known Integral

The integral \( \int \csc^2(\theta) \, d\theta \) is a standard integral that equals \(-\cot(\theta) + C\). Thus,\[\frac{1}{9} \int \csc^2(\theta) \, d\theta = -\frac{1}{9} \cot(\theta) + C.\]

Step 4: Reverse the Trigonometric Substitution

Recall that \( x = 3 \sin(\theta) \), so \( \sin(\theta) = \frac{x}{3} \). The cotangent is \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). Using the Pythagorean identity, \( \cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{x}{3}\right)^2} = \frac{\sqrt{9 - x^2}}{3} \). Therefore, \( \cot(\theta) = \frac{\sqrt{9 - x^2}}{x} \). Substitute back to get:\[-\frac{1}{9} \cot(\theta) = -\frac{1}{9} \cdot \frac{\sqrt{9 - x^2}}{x}.\]

Step 5: Solution

The final solution to the integral is:\[\int \frac{dx}{x^2 \sqrt{9-x^2}} = -\frac{1}{9} \cdot \frac{\sqrt{9 - x^2}}{x} + C.\]

Key Concepts

Definite Integral

A definite integral helps find the accumulation of quantities, such as areas under curves. It is expressed as \[ \int_{a}^{b} f(x) \, dx \]where \( a \) and \( b \) are the limits of integration. These limits define the interval over which you are calculating. The result is a number indicating the total size over this interval.
For instance, if the function \( f(x) \) represents a speed, the definite integral gives the distance covered between the limits \( a \) and \( b \).
Evaluating definite integrals often involves anti-differentiation, where we find a function whose derivative matches the integrand. In this specific problem, even though it's presented as an indefinite integral (no limits), understanding definite integrals can help in comprehending why specific techniques, like substitution, are used to simplify complex integrals.

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \( \theta \), the square of the cosine plus the square of the sine equals one:\[\sin^2(\theta) + \cos^2(\theta) = 1.\]This identity is crucial when performing trigonometric substitutions, especially when the terms inside an integral involve squares of trigonometric functions.
In the exercise, we've seen expressions like \( 9 - x^2 \) being rewritten using a trigonometric substitution. When \( x \) is replaced by a trigonometric function (e.g., \( x = 3 \sin(\theta) \)), it allows the use of the Pythagorean Identity to simplify the expression. More specifically, after the substitution, the expression \( 9 - x^2 \) becomes \( 9 \cos^2(\theta) \), making integration easier.
By simplifying complex algebraic expressions using trigonometric identities, calculus problems become much more manageable.

Integration Techniques

Integration techniques are methods employed to find integrals of functions that aren't easy to integrate directly. Some common techniques include:

• Substitution: Simplifies an integral by changing variables, often using trigonometric or algebraic substitutions.
• Integration by Parts: Decomposes the product of functions into easier parts using the formula \( \int u \, dv = uv - \int v \, du \).
• Partial Fraction Decomposition: Breaks down fractions into simpler components to enable easier integration.

In this specific scenario, trigonometric substitution was used, which is particularly useful when dealing with expressions containing \( \sqrt{a^2 - x^2} \). By choosing \( x = a \sin(\theta) \), the integral turns into a trigonometric one, often easier to tackle with known integrals like \( \int \csc^2(\theta) \, d\theta \).
Understanding how and when to apply these techniques is essential for solving complicated integrals efficiently.