Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(d) ∫₀⁸ 3𝓍(4 ― 𝓍) d(𝓍)

Verified step by step guidance

1. Step 1: Recognize that the integral ∫₀⁸ 3𝓍(4 ― 𝓍) d𝓍 can be split into two parts: ∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍 and ∫₄⁸ 3𝓍(4 ― 𝓍) d𝓍, based on the interval of integration.
2. Step 2: Use the given information that ∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍 = 32 to evaluate the first part of the integral.
3. Step 3: For the second part, ∫₄⁸ 3𝓍(4 ― 𝓍) d𝓍, consider the symmetry of the function 3𝓍(4 ― 𝓍). Analyze whether the function changes sign or remains symmetric over the interval [4, 8].
4. Step 4: If the function is symmetric and the integral over [0, 4] is known, use properties of symmetry to determine the integral over [4, 8]. Alternatively, compute ∫₄⁸ 3𝓍(4 ― 𝓍) d𝓍 directly by substitution or other methods.
5. Step 5: Combine the results of the two integrals, ∫₀⁴ 3𝓍(4 ― 𝓍) d𝓍 and ∫₄⁸ 3𝓍(4 ― 𝓍) d𝓍, to find the value of ∫₀⁸ 3𝓍(4 ― 𝓍) d𝓍.

Key Concepts

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫ₐᵇ f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval [a, b].

Properties of Integrals

The properties of integrals include linearity, additivity, and the ability to change limits. For instance, the integral of a sum is the sum of the integrals, and the integral from a to b can be expressed as the negative of the integral from b to a. These properties allow for simplification and manipulation of integrals to facilitate evaluation.

Substitution Method

The substitution method is a technique used to simplify the evaluation of integrals by changing the variable of integration. By substituting a new variable, often denoted as u, the integral can be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand can be expressed in terms of a simpler function.