91Ó°ÊÓ

Integral evaluation is a fundamental process in calculus, where the goal is to find the integral or antiderivative of a function. When evaluating integrals, it’s important to recognize the form of the integral to select the most suitable technique. In our case, we have the integral \( \int \sqrt{1-9t^2} \, dt \). This integral represents a square root function inside the integrand, which often hints at the need for a specialized method like trigonometric substitution.

Evaluating such integrals generally involves the following steps:

• Identify the structure of the integrand. If it matches forms involving \( a^2 - x^2 \), \( a^2 + x^2 \), or \( x^2 - a^2 \), consider using trigonometric substitutions.
• Use appropriate trigonometric identities to simplify the integral into a more easily integrable form.
• Perform the integration, which may include using identities and simplifying expressions.
• Formally substitute back to the original variable to express the solution in terms of the initial variable.

These steps ensure the integral is evaluated systematically, transforming a complex problem into a simpler one.

Trigonometric identities play a pivotal role when solving integrals that involve trigonometric functions. They allow us to simplify expressions, making them much easier to integrate. In this exercise, we encountered the integral with a \( \cos^2(\theta) \) term due to the substitution process.

To manage this term, one of the most useful identities is:

• \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \)

This identity transforms the squared cosine function into a form that separates constant and trigonometric components, making integration straightforward. By using this identity:

• The integral was simplified from \( \int \cos^2(\theta) \, d\theta \) to \( \frac{1}{6} \int (1 + \cos(2\theta)) \, d\theta \).

This resulted in a more digestible problem that can be solved term by term, handling constants and simpler trigonometric integrals separately.

The substitution method is a classical technique in calculus used to evaluate integrals, particularly when dealing with functions that involve compositions. It is particularly powerful when the integrand includes expressions of the form \( a^2 - x^2 \), allowing the use of trigonometric substitutions.

In this problem, we employed trigonometric substitution, choosing \( t = \frac{1}{3} \sin(\theta) \). This choice transforms the integrand's structure, replacing \( t \) with a trigonometric function. The process involved the following:

• Recognizing that \( \sqrt{1 - 9t^2} \) can be transformed using \( t = \frac{1}{3} \sin(\theta) \).
• Finding \( dt \) in terms of \( \theta \) by differentiating \( t = \frac{1}{3} \sin(\theta) \), giving \( dt = \frac{1}{3} \cos(\theta) \, d\theta \).
• Substituting \( t \) and \( dt \) into the integral, converting the variable from \( t \) to \( \theta \).
• Integrating using transformed expressions, then substituting back the original variable post-integration.

Such substitution rewrites the integral with more tractable trigonometric functions, greatly simplifying the evaluation process. This method not only simplifies the integration task but also connects trigonometric functions with polynomial expressions elegantly.