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.
$$
\int_{1}^{3} \frac{2 x^{2}+3 x-2}{x} d x
$$

Key Concepts

Definite Integration

Definite integration involves calculating the accumulation of quantities over a specified interval. It uses the antiderivative of the function combined with the evaluation at the boundaries according to the Fundamental Theorem of Calculus. This process results in a numerical value representing the net area under a curve between two points.

Algebraic Simplification

Before applying integration techniques, it is often useful to simplify the given integrand. In cases where the integrand includes a rational expression, algebraic manipulation such as polynomial division or splitting the fraction into simpler terms can transform a complex expression into a sum of terms that are easier to integrate.

Fundamental Theorem of Calculus

This theorem bridges the concepts of differentiation and integration by stating that the definite integral of a function over an interval is obtained by finding an antiderivative of the function and evaluating it at the endpoints of the interval. It provides the justification for taking the difference of the antiderivative values at the upper and lower limits to compute the integral.

Graphing Utility Verification

Using a graphing utility or computer algebra system to verify the result of an integration exercise is an effective way to check the accuracy of manual calculations. This technology can plot the function and numerically evaluate the integral, providing a useful visual and computational confirmation of the theoretical result.

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