Question

Let f be a function such that f(-x) = -f(x) for all x. If ∫[0 to 2] f(x) dx = 5, then ∫[-2 to 2] (f(x) + 6) dx =

(A) 6
(B) 16
(C) 24
(D) 34

Answer 1

The integral of the odd function f(x) from -2 to 2 cancels out, leaving only the integral of the constant 6, which when evaluated from -2 to 2 is 24. Therefore, the integral sought is 24.

Step-by-step explanation:

The student's question pertains to properties of odd and even functions and how they apply to definite integrals. Specifically, the function f mentioned in the question is an odd function because it satisfies the condition f(-x) = -f(x) for all x.

Firstly, it's given that ∫[0 to 2] f(x) dx = 5. This integral represents the area under the curve of f(x) from 0 to 2. However, since f(x) is an odd function, the integral from -2 to 0 of f(x) would yield an area of -5, because the areas on the negative side of the y-axis are the equal in magnitude but opposite in sign to the areas on the positive side.

Next, we are asked to determine the value of ∫[-2 to 2] (f(x) + 6) dx. We can split this integral into two parts: ∫[-2 to 0] (f(x) + 6) dx + ∫[0 to 2] (f(x) + 6) dx. The integral of f(x) from -2 to 0 is -5 (from the odd function property), and the integral from 0 to 2 of f(x) is 5 (as given). Integrating 6 over the interval [-2, 2] simply yields 6 times the length of the interval, which is 24.

Combining these results, the first part (-5) cancels out with the f(x) from 0 to 2 (5), and we are left with the integral of 6, which is 24. Hence, ∫[-2 to 2] (f(x) + 6) dx = 24, or answer choice (C).

Answer 2

Using the properties of an odd function and knowing the integral of f(x) from 0 to 2, the integral of (f(x) + 6) from -2 to 2 is calculated to be 24, which is option (C).

Step-by-step explanation:

The question deals with the properties of an odd function, particularly in the context of integration. An odd function such as xe-x² satisfies the condition that f(-x) = -f(x) for all x in its domain. When integrating an odd function over an interval symmetric about the origin, the result is zero, since the positive and negative areas cancel each other out.

Given that the integral of f(x) from 0 to 2 is 5, we can infer that the integral from -2 to 0 would be -5 due to the odd function property. Hence, the integral of f(x) from -2 to 2 is 0. Adding 6 to f(x) and integrating results in just integrating 6 over the interval from -2 to 2, which equals 6 times the length of the interval, 4, thus arriving at 24 as our answer.

So, to find the integral of f(x) + 6 from -2 to 2, we add the areas: 0 (from f(x)) + 6 * 4 = 24, which corresponds to choice (C).