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.
$$
\int_{0}^{1} x e^{-x^{2}} d x
$$

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Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with that of integration. It states that if a function is continuous on a closed interval, then its definite integral over that interval can be found using any of its antiderivatives. This theorem forms the backbone of evaluating definite integrals in calculus.

u-Substitution

The u-substitution method is a technique for simplifying the process of integration by substituting a part of the integrand with a new variable 'u'. This method converts the integral into a simpler form by accounting for the derivative of the substitution, thereby making it easier to integrate functions, especially when dealing with compositions of functions.

Definite Integral

A definite integral represents the accumulation of quantities, often interpreted as the area under a curve between two specified limits. It provides an exact value by evaluating an antiderivative at the upper limit and subtracting its value at the lower limit, which is a central concept in calculus.

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