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.                                                                                                                                                           
                                                                                                                                                                    
(a) ∫ ƒ(𝓍) ƒ'(𝓍) d𝓍 = ½ (ƒ(𝓍))² + C.

Verified step by step guidance

1. Step 1: Recall the integration by substitution method. To evaluate ∫ ƒ(𝓍) ƒ'(𝓍) d𝓍, consider substituting u = ƒ(𝓍). This substitution simplifies the integral.
2. Step 2: Compute the derivative of u with respect to 𝓍. Since u = ƒ(𝓍), then du/d𝓍 = ƒ'(𝓍), or equivalently, du = ƒ'(𝓍) d𝓍.
3. Step 3: Rewrite the integral in terms of u. Substituting u = ƒ(𝓍) and du = ƒ'(𝓍) d𝓍, the integral becomes ∫ u du.
4. Step 4: Solve the integral ∫ u du. The antiderivative of u with respect to u is (1/2)u² + C, where C is the constant of integration.
5. Step 5: Substitute back u = ƒ(𝓍) into the result. This gives (1/2)(ƒ(𝓍))² + C, which matches the given statement. Therefore, the statement is true.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval equals the change in the function's values. This theorem is crucial for evaluating definite integrals and understanding the relationship between a function and its antiderivative.

Integration by Substitution

Integration by substitution is a technique used to simplify the process of integration by changing the variable of integration. It involves substituting a part of the integrand with a new variable, which can make the integral easier to solve. This method is particularly useful when dealing with composite functions.

Continuous Functions

A function is continuous if it does not have any breaks, jumps, or holes in its graph. For the statements in the question, the continuity of the functions ƒ, ƒ', and ƒ'' ensures that the properties of limits and integrals apply, allowing for the application of the Fundamental Theorem of Calculus and ensuring that the results derived from these functions are valid.