91Ó°ÊÓ

Integrals are a fundamental concept in calculus that allow us to determine the accumulation of quantities, like area under a curve or total change over an interval. In this problem, we are tasked with solving an integral using trigonometric substitution, a technique used when direct integration is challenging due to complex expressions.

Given the function \( \int \frac{d x}{x^{2} \sqrt{4-x^{2}}} \), our goal is to find an antiderivative, a function whose derivative gives us back the original function inside the integral sign. This requires careful substitutions and simplifications to rewrite the original integral into a form that's easier to integrate.

The substitution \( x = 2 \sin \theta \) transforms our integral into the world of trigonometric calculus, where we can utilize identities to simplify and solve.

Trigonometric identities play a crucial role in simplifying expressions during integration, particularly when employing substitution techniques. In this exercise, the substitution \( x = 2 \sin \theta \) simplifies our radical expression \( \sqrt{4-x^2} \) by leveraging identities.

Here's how it works step-by-step:

• The expression \( 4 - x^2 \) becomes \( 4 - (2 \sin \theta)^2 = 4 \cos^2 \theta \) by using the Pythagorean identity \( 1 - \sin^2 \theta = \cos^2 \theta \).
• Substituting these simplifications back into the integral enables us to express it in terms of trigonometric functions, which are often easier to integrate.

Understanding and applying these identities allows us to transform and simplify complicated integrals into a more manageable form.

The use of a right triangle in trigonometric substitution is a powerful visualization aid that simplifies the conceptual understanding of these problems. We can sketch a right triangle to relate the given trigonometric substitution and better grasp the relationships between sides and the angle \( \theta \).

For substitution \( x = 2 \sin \theta \), we sketch a right triangle where:

• The hypotenuse measures 2 (because \( \sin \theta = \frac{x}{2} \))
• The opposite side measures \( x \)
• The adjacent side is \( \sqrt{4-x^2} \)

The triangle helps us visualize and compute trigonometric functions such as \( \cot \theta \), which is used in back-substituting \( \theta \) to terms of \( x \) into the antiderivative solution.

Finding an antiderivative means determining a function whose derivative returns to the integrand. In our problem, after simplifying and substituting, we arrive at the integral \( \int \csc^2 \theta \, d\theta \).

To find the antiderivative, we use trigonometric integration rules:

• The antiderivative of \( \csc^2 \theta \) is \( -\cot \theta \).
• Thus, the solution in terms of \( \theta \) becomes \( -\frac{1}{4} \cot \theta + C \).

The final step involves back-substituting the trigonometric expression for the original variable \( x \). Using the right triangle relationships, \( \cot \theta \) resolves to \( \frac{\sqrt{4-x^2}}{x} \), providing the complete solution in terms of \( x \).

This showcases the elegance of trigonometric substitution: simplifying complex expressions and solving challenging integrals by transforming them into familiar forms.