Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.                                                                                                                                                           
                                                                                                                                                                    
(c) ∫ sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 d𝓍 .

Verified step by step guidance

1. Step 1: Begin by analyzing the given statement: ∫ sin(2𝓍) d𝓍 = 2 ∫ sin(𝓍) d𝓍. The goal is to determine whether this equality holds true or not.
2. Step 2: Recall the property of definite integrals and substitution. The integral of a function multiplied by a constant inside the argument does not directly equate to multiplying the integral by that constant. Instead, substitution is required to evaluate the integral properly.
3. Step 3: Use substitution to evaluate ∫ sin(2𝓍) d𝓍. Let u = 2𝓍, which implies du = 2 d𝓍. Rewrite the integral as (1/2) ∫ sin(u) du.
4. Step 4: Compare the rewritten integral (1/2) ∫ sin(u) du with the original statement. Notice that the factor of 1/2 appears due to the substitution, which contradicts the claim that ∫ sin(2𝓍) d𝓍 = 2 ∫ sin(𝓍) d𝓍.
5. Step 5: Conclude that the given statement is false. Provide a counterexample or explanation showing that the integral of sin(2𝓍) is not simply twice the integral of sin(𝓍), but rather involves a scaling factor due to substitution.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It is the reverse process of differentiation and can be used to calculate quantities such as total distance, area, and volume. Understanding the properties of integrals, including linearity and substitution, is crucial for solving integration problems.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In calculus, these functions are essential for modeling oscillatory behavior and are frequently encountered in integration and differentiation. Recognizing the properties and identities of trigonometric functions is vital for simplifying expressions and solving integrals involving these functions.

Continuous Functions

A continuous function is one that does not have any breaks, jumps, or holes in its graph. For a function to be continuous at a point, the limit of the function as it approaches that point must equal the function's value at that point. In the context of the question, the continuity of the functions ƒ, ƒ', and ƒ'' ensures that the properties of integration and differentiation can be applied without concern for discontinuities, which is essential for validating the given statement.