Evaluating a Definite Integral In Exercises 57-64 , evaluate...

Evaluating a Definite Integral In Exercises $57-64$ , evaluate the definite integral. Use the integration capabilities of a graphing utility to verify your result.
$$
\int_{0}^{2 / \sqrt{3}} \frac{1}{4+9 x^{2}} d x
$$

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Key Concepts

Definite Integral

A definite integral is the evaluation of the integral of a function over a specified interval. It represents the net area under the curve of the function between two bounds and is calculated by first finding an antiderivative and then applying the Fundamental Theorem of Calculus using the limits of integration.

Substitution Method

The substitution method is a technique used to simplify the process of integration by changing variables. By substituting part of the integrand with a new variable, the integral often transforms into a form that is easier to integrate, and the limits of integration must be adjusted accordingly if the integral is definite.

Standard Integral of the Form 1/(a^2 + x^2)

Integrals that have the form 1/(a^2 + x^2) are commonly integrated using the standard arctangent formula, which states that ?dx/(a^2 + x^2) equals (1/a) arctan(x/a) + C. Recognizing this form helps in quickly solving integrals by identifying the appropriate substitution and constants.

Graphing Utility Verification

Using a graphing utility to verify an evaluated integral involves plotting the function or using built-in numerical integration features to calculate the value of the definite integral. This computational verification complements analytical techniques by confirming the accuracy of the obtained result.

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