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.                                                                                                                                                           
                                                                                                                                                                    
(b) ∫ (ƒ(𝓍))ⁿ ƒ'(𝓍) d𝓍 = 1/(n + 1) (ƒ(𝓍))ⁿ⁺¹ + C , n ≠ ―1 .

Verified step by step guidance

1. Step 1: Recognize that the problem involves verifying the given integral formula. The formula states that the integral of (ƒ(𝓍))ⁿ multiplied by ƒ'(𝓍) with respect to 𝓍 equals 1/(n + 1) * (ƒ(𝓍))ⁿ⁺¹ + C, where n ≠ -1.
2. Step 2: Use substitution to simplify the integral. Let u = ƒ(𝓍). Then, du = ƒ'(𝓍) d𝓍. This substitution transforms the integral into ∫ uⁿ du.
3. Step 3: Recall the power rule for integration. The integral of uⁿ with respect to u is (1/(n + 1)) * uⁿ⁺¹ + C, provided n ≠ -1. This matches the structure of the given formula.
4. Step 4: Substitute back u = ƒ(𝓍) into the result of the integration. This gives (1/(n + 1)) * (ƒ(𝓍))ⁿ⁺¹ + C, which is consistent with the formula provided in the problem.
5. Step 5: Conclude that the formula is correct for n ≠ -1, as the derivation using substitution and the power rule confirms its validity. The condition n ≠ -1 ensures the denominator (n + 1) does not become zero, which would make the formula undefined.

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 at the endpoints. 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. It involves changing the variable of integration to make the integral easier to solve. This method is particularly useful when dealing with composite functions, as it allows us to express the integral in terms of a single variable, making it more manageable.

Continuous Functions

A function is considered continuous if there are no 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. Continuity is essential in calculus because many theorems, including the Fundamental Theorem of Calculus, require the functions involved to be continuous over the interval of integration.