Evaluate Definite Integral Using Fundamental Theorem of Calculus Part 2

Evaluate the integral from 0 to 2 of $\displaystyle \int_{0}^{2} (5x^4 - 2x) \, dx$ using the fundamental theorem of calculus Part 2.

The problem of evaluating the integral from zero to two of the polynomial function $5x^4 - 2x$ using the fundamental theorem of calculus part 2 is an essential exercise for mastering calculus. The fundamental theorem of calculus connects differentiation and integration, providing a powerful method to calculate definite integrals. It states that if a function is continuous on a closed interval and has an antiderivative, then the definite integral over that interval is equal to the difference of the values of the antiderivative at the endpoints.

This problem involves finding an antiderivative of the given polynomial function. Once you determine the antiderivative, you evaluate it at the upper and lower limits of the integral and subtract these values to find the result. Polynomials are among the simpler types of functions to integrate, and this particular problem provides an opportunity to practice the application of the fundamental theorem in a straightforward context, reinforcing the foundational skills necessary for more complex integration problems.

Understanding this concept deeply is crucial as it forms the basis for tackling more advanced integral calculations often encountered in real-world applications and further mathematical studies. Grasping the fundamental theorem not only aids in evaluating integrals but also enhances comprehension of the relationship between the accumulation of quantities and the rate of change, which is a cornerstone of calculus.

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