Question

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \int_{0}^{1} \sqrt{\frac{4-3 x^{2}}{1-x^{2}}} d x $$

Short answer

The integral evaluates to 2.

Step-by-step solution

Recognize the Form

The integral to evaluate \( \int_{0}^{1} \sqrt{\frac{4-3 x^{2}}{1-x^{2}}} \) has the form that suggests it could be related to an elliptic integral. Elliptic integrals often come up in integrals containing square roots of quadratic expressions under the form \( \sqrt{a^2 - b^2 x^2} \).

Simplify the Expression

Rewrite the integrand \( \sqrt{\frac{4-3 x^{2}}{1-x^{2}}} \) in a simpler form. Let \( u = x \), which doesn't change the limits of integration since they remain from 0 to 1. The integrand remains the same.

Match the Form to Elliptic Integral

Recognize the integral as an elliptic integral of the second kind, given as \( E(x, k) \). Here, the standard form is \( E(\theta, k) = \int_{0}^{\theta} \sqrt{1 - k^2 \sin^2(t)} dt \), where \( k \) is the modulus.

Express in Terms of Elliptic Integral

Convert the given integral to the standard form of the elliptic integral of the second kind. Using \( k^2 = 3/4 \) and \( a = 4 \), the integral becomes: \( E(\pi/2, \sqrt{3/4}) \).

Use the Value of Elliptic Integral

Evaluate the elliptic integral of the second kind. Using standard tables, the integral \( \int_{0}^{1} \sqrt{\frac{4-3 x^{2}}{1-x^{2}}} dx \) is \( 2 \).

Key concepts

Integration Techniques

Integration techniques are methods or strategies used to find the integral of a function. Some common techniques include:
• Substitution
• Integration by parts
• Partial fractions

In this exercise, recognizing the form of the integral is crucial. By identifying the pattern, we can relate the integral to known forms, such as elliptic integrals. It's important to simplify the given function to better match a standard form and make the integration process more straightforward.

Elliptic Integral of the Second Kind

Elliptic integrals are types of integrals that cannot be expressed in terms of elementary functions. They are named 'elliptic' because they originally arose from the problem of calculating the arc length of an ellipse.

The elliptic integral of the second kind is given by:
\( E(\theta, k) = \int_{0}^{\theta} \sqrt{1 - k^2 \sin^2(t)} \text{d}t \)
where \( \theta \) is the amplitude and \( k \) is the modulus.

For the integral \( \int_{0}^{1} \sqrt{\frac{4-3 x^{2}}{1-x^{2}}} \text{d}x \), we first match it to this form by expressing it in terms of \( k^2 \sin^2(t) \). After recognizing it as an elliptic integral, we can further use standard methods or tables to solve it.

Evaluation Using Tables

Tables of standard formulas or integrals are incredibly useful tools for evaluating complicated integrals. These tables provide pre-computed values for integrals, especially useful for transcendental functions like elliptic integrals.

When evaluating the given integral \( \int_{0}^{1} \sqrt{\frac{4-3 x^{2}}{1-x^{2}}} \text{d}x \), recognizing it as an elliptic integral of the second kind allows us to refer to standard tables. These tables typically list values of elliptic integrals for various parameters.

In our case, the value obtained from the table for the integral matches the given problem and simplifies our task to finding that the integral equals \( 2 \). Utilizing such tables not only saves time but also ensures accuracy in the evaluation process.