Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       
(a) ∫¹₋₁ 𝓍ƒ(𝓍²) d𝓍

Verified step by step guidance

1. Step 1: Recognize that the given function ƒ is even, meaning ƒ(𝓍) = ƒ(-𝓍). This property will be useful in simplifying the integral.
2. Step 2: Analyze the integral ∫₋₁¹ 𝓍ƒ(𝓍²) d𝓍. Notice that the integrand contains the term 𝓍, which is an odd function (𝓍 = -𝓍 when reflected about the origin).
3. Step 3: Recall the property of definite integrals: If the integrand is an odd function and the limits of integration are symmetric about zero (e.g., from -a to a), then the integral evaluates to 0. In this case, 𝓍ƒ(𝓍²) is an odd function because 𝓍 is odd and ƒ(𝓍²) is even.
4. Step 4: Conclude that the integral ∫₋₁¹ 𝓍ƒ(𝓍²) d𝓍 = 0 due to the symmetry of the integrand and the limits of integration.
5. Step 5: No further computation is needed because the integral evaluates to 0 based on the properties of odd and even functions.

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 means that the integral of an even function over a symmetric interval, such as [-a, a], can be simplified. For example, if f is even, then ∫₋ₐ⁺ₐ f(x) dx = 2∫₀⁺ₐ f(x) dx.

Substitution in Integrals

Substitution is a technique used in integration to simplify the process of finding an integral. It involves changing the variable of integration to make the integral easier to evaluate. For instance, if we let u = g(x), then the integral ∫ f(g(x)) g'(x) dx can be transformed into ∫ f(u) du, which may be simpler to solve.

Definite Integrals

A definite integral represents the signed area under a curve defined by a function f(x) between two limits, a and b. It is denoted as ∫ₐᵇ f(x) dx and can be computed using the Fundamental Theorem of Calculus, which states that if F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) - F(a). This concept is crucial for evaluating integrals over specific intervals.