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.
$$
\int_{0}^{\pi / 4} \cos 2 x d x
$$

Key Concepts

Definite Integral

A definite integral represents the accumulation of quantities, such as area, over a specific interval. It calculates the net signed area between the function's graph and the horizontal axis from one endpoint to another.

Fundamental Theorem of Calculus

This theorem bridges differentiation and integration, stating that if a function is continuous on a closed interval, then its definite integral over that interval can be computed using its antiderivative. Specifically, the difference between the values of the antiderivative at the endpoints gives the value of the integral.

Antiderivative

An antiderivative of a function is another function whose derivative equals the original function. In the context of definite integrals, finding an antiderivative is key because it allows one to apply the Fundamental Theorem of Calculus to evaluate the integral over a given interval.

U-Substitution

U-substitution is a technique used to simplify the integration of composite functions by substituting part of the integrand with a new variable. This method often converts an integral into a more familiar form, making it easier to find an antiderivative.

Graphing Utility in Integration

Graphing utilities, such as graphing calculators or computer algebra systems, can perform numerical or symbolic integration. They help verify analytical results by providing an independent computation of the integral, ensuring the accuracy of the evaluated value.

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