63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If m is a positive integer, then ∫[0 to π] cos^(2m+1)(x) dx = 0.

Verified step by step guidance

1. Step 1: Begin by analyzing the integral ∫[0 to π] cos^(2m+1)(x) dx. The integrand is cos^(2m+1)(x), which is an odd power of the cosine function. Recall that cosine is an even function, meaning cos(-x) = cos(x). However, raising cosine to an odd power makes the overall function odd, as odd powers of cosine reverse the sign when x is replaced with -x.
2. Step 2: Recall the property of definite integrals for odd functions. If f(x) is an odd function and the limits of integration are symmetric about zero (e.g., from -a to a), then ∫[-a to a] f(x) dx = 0. However, in this case, the limits of integration are from 0 to π, which are not symmetric about zero. This means the odd function property does not directly apply here.
3. Step 3: To determine whether the integral evaluates to zero, consider the behavior of cos^(2m+1)(x) over the interval [0, π]. The cosine function is positive on [0, π/2] and negative on [π/2, π]. Raising cosine to an odd power preserves the sign of the function, meaning cos^(2m+1)(x) is positive on [0, π/2] and negative on [π/2, π].
4. Step 4: Split the integral into two parts: ∫[0 to π/2] cos^(2m+1)(x) dx and ∫[π/2 to π] cos^(2m+1)(x) dx. The first integral represents the positive contribution, while the second integral represents the negative contribution. However, these contributions do not necessarily cancel out because the intervals [0, π/2] and [π/2, π] are not symmetric, and the magnitude of the function may differ across these intervals.
5. Step 5: Conclude that the statement 'If m is a positive integer, then ∫[0 to π] cos^(2m+1)(x) dx = 0' is false. Provide a counterexample by evaluating the integral for a specific value of m (e.g., m = 1) to show that the integral does not equal zero. This demonstrates that the odd power of cosine does not lead to cancellation over the interval [0, π].

Key Concepts

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. In this case, the integral of cos^(2m+1)(x) from 0 to π is evaluated to determine if it equals zero. Understanding how to compute definite integrals and the properties of the integrand is crucial for solving the problem.

Properties of the Cosine Function

The cosine function is periodic and symmetric, specifically even, meaning cos(-x) = cos(x). For odd powers of cosine, such as cos^(2m+1)(x), the function exhibits symmetry about the y-axis, which can lead to cancellation of areas under the curve over symmetric intervals. This property is essential for determining the value of the integral.

Odd and Even Functions

An odd function satisfies the condition f(-x) = -f(x), while an even function satisfies f(-x) = f(x). The function cos^(2m+1)(x) is odd because it is raised to an odd power. When integrating an odd function over a symmetric interval like [0, π], the result is zero, which is key to answering the question.