Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus


∫₋₂⁻¹ 𝓍⁻³ d𝓍

Verified step by step guidance

1. Step 1: Recall the Fundamental Theorem of Calculus, which states that if a function f(x) is continuous on [a, b] and F(x) is its antiderivative, then ∫ₐᵇ f(x) dx = F(b) - F(a).
2. Step 2: Identify the integrand, which is x⁻³ (or 1/x³). The goal is to find its antiderivative. Rewrite the integrand as x⁻³ for clarity.
3. Step 3: Compute the antiderivative of x⁻³. Using the power rule for integration, ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1. For x⁻³, n = -3, so the antiderivative becomes -1/(2x²).
4. Step 4: Apply the limits of integration, -2 and -1, to the antiderivative. Substitute x = -1 and x = -2 into the antiderivative expression, F(x) = -1/(2x²).
5. Step 5: Calculate the difference F(-1) - F(-2) to evaluate the definite integral. This step involves substituting the values and simplifying the result.

Key Concepts

Definite Integrals

Definite integrals represent the signed area under a curve between two specified limits on the x-axis. They are denoted as ∫[a, b] f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The value of a definite integral provides a numerical result that quantifies the accumulation of quantities, such as area, over the interval [a, b].

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with 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 and calculating its values at the limits of integration.

Antiderivatives

An antiderivative of a function f(x) is another function F(x) such that F'(x) = f(x). Finding an antiderivative is essential for evaluating definite integrals using the Fundamental Theorem of Calculus. For example, if f(x) = x^n, the antiderivative is F(x) = (x^(n+1))/(n+1) + C, where C is a constant. Understanding how to find antiderivatives is crucial for solving integral problems.