Evaluating integrals Evaluate the following integrals.


∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral ∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍 is a definite integral, meaning we will evaluate the antiderivative of the function and then compute the difference between its values at the upper and lower limits.
2. Step 2: Break the integral into separate terms for easier computation: ∫₋₂² (3𝓍⁴) d𝓍 - ∫₋₂² (2𝓍) d𝓍 + ∫₋₂² (1) d𝓍.
3. Step 3: Compute the antiderivative of each term: For 3𝓍⁴, the antiderivative is (3/5)𝓍⁵; for -2𝓍, the antiderivative is -𝓍²; and for 1, the antiderivative is 𝓍.
4. Step 4: Apply the Fundamental Theorem of Calculus: Substitute the upper limit (𝓍 = 2) and lower limit (𝓍 = -2) into the antiderivative of each term, and compute the difference between the values at these limits.
5. Step 5: Combine the results from each term to find the total value of the definite integral. This involves adding the contributions from (3/5)𝓍⁵, -𝓍², and 𝓍 after evaluating them at the limits.

Key Concepts

Definite Integrals

A definite integral calculates the accumulation of a function's values over a specific interval, represented as ∫[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.

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 ∫[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.

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 3x⁴ - 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.