Suppose ƒ is an even function and ∫⁸₋₈ ƒ(𝓍) d𝓍 = 18
(b) Evaluate ∫₋₈⁸ 𝓍ƒ(𝓍) d𝓍 .

Verified step by step guidance

1. Step 1: Recall the definition of an even function. An even function satisfies ƒ(𝓍) = ƒ(-𝓍) for all 𝓍 in its domain. This symmetry will be important for solving the integral.
2. Step 2: Consider the integral ∫₋₈⁸ 𝓍ƒ(𝓍) d𝓍. Notice that the integrand 𝓍ƒ(𝓍) involves the product of 𝓍 and ƒ(𝓍).
3. Step 3: Analyze the symmetry of the integrand. Since ƒ(𝓍) is even, ƒ(𝓍) = ƒ(-𝓍). However, 𝓍 is an odd function because 𝓍 = -𝓍 when reflected about the origin. The product of an even function and an odd function is an odd function.
4. Step 4: Recall a key property of definite integrals: the integral of an odd function over a symmetric interval [−a, a] is always 0. This is because the positive and negative contributions cancel each other out.
5. Step 5: Conclude that ∫₋₈⁸ 𝓍ƒ(𝓍) d𝓍 = 0 based on the symmetry of the integrand and the properties of odd functions over symmetric intervals.

Key Concepts

Even Functions

An even function is defined as a function f(x) that satisfies the condition f(-x) = f(x) for all x in its domain. This symmetry about the y-axis implies that the area under the curve from -a to 0 is equal to the area from 0 to a. This property is crucial for evaluating integrals involving even functions, as it simplifies calculations and allows for certain symmetries to be exploited.

Properties of Definite Integrals

Definite integrals have several important properties, one of which is that the integral of an odd function over a symmetric interval around zero is zero. This is because the areas above and below the x-axis cancel each other out. Understanding these properties helps in evaluating integrals more efficiently, especially when dealing with functions that exhibit symmetry.

Integration by Parts

Integration by parts is a technique used to integrate products of functions and is based on the product rule for differentiation. It is expressed as ∫u dv = uv - ∫v du, where u and v are differentiable functions. This method is particularly useful when integrating products of polynomials and other functions, allowing for the transformation of complex integrals into simpler forms.