Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the accumulation of a function's values over a specific interval, represented as β«[a,b] f(x) dx. The result is a numerical value that represents the area under the curve of the function f(x) from x = a to x = b. Understanding the limits of integration and how they affect the area calculation is crucial for evaluating definite integrals.
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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then β«[a,b] f(x) dx = F(b) - F(a). This theorem allows us to evaluate definite integrals by finding the antiderivative of the integrand, simplifying the process of calculating areas under curves.
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Polynomial Functions
Polynomial functions are expressions of the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n are coefficients and n is a non-negative integer. In the given integral, the function 3xβ΄ - 2x + 1 is a polynomial, and understanding how to integrate polynomial functions is essential, as they can be integrated term by term using the power rule.
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