Problem 6 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-14\) $$ \int_{0}^{1 / 2 \sqrt{2}} \frac{2 d x}{\sqrt{1-4 x^{2}}} $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{\pi}{4} \).

Step 1: Identify the Integral Type

The integral we are dealing with is \( \int \frac{2 \, dx}{\sqrt{1 - 4x^2}} \). This is similar to the integral form \( \frac{du}{\sqrt{a^2 - u^2}} \), which corresponds to an inverse trigonometric function.

Step 2: Perform a Trigonometric Substitution

To handle \( \sqrt{1 - 4x^2} \), use the substitution \( x = \frac{1}{2} \sin{\theta} \), which means \( dx = \frac{1}{2} \cos{\theta} \, d\theta \). Then the limit of integration changes: when \( x = 0, \theta = 0 \) and when \( x = \frac{1}{2\sqrt{2}}, \theta = \frac{\pi}{4} \).

Step 3: Substitute and Simplify the Integral

Substitute \( x = \frac{1}{2} \sin{\theta} \) into the integral,\[\int_{0}^{\pi/4} \frac{2 \cdot \frac{1}{2} \cos{\theta} \, d\theta}{\sqrt{1 - 4 \left(\frac{1}{2} \sin{\theta}\right)^2}} = \int_{0}^{\pi/4} \frac{\cos{\theta} \, d\theta}{\cos{\theta}} = \int_{0}^{\pi/4} d\theta.\]

Step 4: Evaluate the Simplified Integral

Now integrate with respect to \( \theta \):\[\int_{0}^{\pi/4} d\theta = \left[ \theta \right]_{0}^{\pi/4} = \frac{\pi}{4} - 0.\]

Step 5: Provide the Final Answer

The value of the original integral \( \int_{0}^{1 / 2 \sqrt{2}} \frac{2 \, dx}{\sqrt{1 - 4x^2}} \) is \( \frac{\pi}{4} \).

Key Concepts

Integral Evaluation

Integral evaluation is the process of finding the value of an integral, which essentially means calculating the area under a curve or solving a specific problem described by an integral. In this exercise, we were asked to evaluate the integral \( \int_{0}^{1 / 2 \sqrt{2}} \frac{2 \, dx}{\sqrt{1-4 x^{2}}} \). This type of integral initially appears complex due to the radical in the denominator. Such integrals may involve transforming the integral into a more manageable form, enabling us to utilize known techniques and methods.

• Start by recognizing the form of the integral. Is it related to a standard or special function?
• Decide if any substitutions can simplify the integral.
• Apply standard techniques to evaluate it.

In this case, identifying the presence of a function reminiscent of \, \( \frac{du}{\sqrt{a^2 - u^2}} \), suggests the use of inverse trigonometric functions, paving the way for trigonometric substitution, which simplifies the process drastically.

Inverse Trigonometric Functions

Inverse trigonometric functions arise when dealing with integrals that resemble the integral forms tied to these functions. Typical examples include expressions like \( \sqrt{a^2 - x^2} \) in the denominator. The integral from the exercise resembles these inverse trigonometric function forms, which imply that the solution aligns with the arcsin function due to the presence of a square root in the form \( \sqrt{1 - 4x^2} \). Subbing the expression with \( x = \frac{1}{2} \sin{\theta} \) effectively transforms it into a recognizable form.Inverse trigonometric functions, like \( \arcsin(x) \), are essential when dealing with such integrals, as they simplify otherwise cumbersome expressions, ultimately leading us to straightforward results after substitution and evaluating the integral.

Integration Techniques

Integration techniques are methods used to solve integrals, especially when they are not straightforward to evaluate. In this exercise, the technique used was trigonometric substitution. This substitution method is beneficial when the integral contains an expression like \( \sqrt{a^2 - x^2} \), allowing us to replace \( x \) with a trigonometric function of a new variable like \( \theta \). Here's a breakdown of the trigonometric substitution process:

• Identify a suitable substitution: Choose \( x = \frac{a}{b} \sin{\theta} \) for \( \sqrt{a^2 - x^2} \) forms, where \( a^2 = 1 \) and \( b = 2 \) in this case.
• Differentiate your substitution to find \( dx \), e.g., \( dx = \frac{1}{2} \cos{\theta} \, d\theta \).
• Change the limits of integration based on the substitution, converting \( x \)-limits to \( \theta \)-limits.
• Simplify and evaluate the integral with respect to \( \theta \).

This effectively transforms a complex integral into a simpler one, often leading to an easier or known solution form, which can then be integrated with confidence.