Questions: Antiderivatives

Question 1

What is the antiderivative of f(x) = 4x³?

• A. 12x² + C
• B. x⁴ + C
• C. x⁴/3 + C
• D. x⁴

Answer: (B)

Applying the power rule in reverse: increase the exponent from 3 to 4, then divide by the new exponent (4), giving 4x⁴/4 = x⁴. The +C is required because any constant added to F(x) also satisfies F'(x) = f(x). Option A applies the *derivative* rule (multiply by exponent, decrease exponent) instead of reversing it. Option C divides by the original exponent (3) rather than the new one (4). Option D is correct but omits the necessary +C.

Question 2

Student A claims an antiderivative of 2x is x² + 3. Student B claims it is x² − 7. Who is correct?

• A. Student A only — the constant 3 matches the original function's offset
• B. Student B only — antiderivatives cannot have positive added constants
• C. Both are correct — antiderivatives of the same function form a family differing by a constant
• D. Neither — the only correct antiderivative of 2x is x²

Answer: (C)

Both x² + 3 and x² − 7 differentiate to 2x, because the derivative of any constant is zero. The *general* antiderivative is x² + C, where C can be any real number. There is no unique antiderivative without additional information (such as a specific point the function must pass through). Option D is the most common mistake: writing just x² is one valid antiderivative, but it is not the only one.

Question 3

An antiderivative of a function is unique — for any given f(x), there is exactly one F(x) such that F'(x) = f(x).

• T. True
• F. False

Answer: False

Antiderivatives are never unique. Because the derivative of any constant is zero, if F(x) is an antiderivative of f(x), then so is F(x) + C for any constant C. This yields an entire family of antiderivatives, all shifted vertically from one another. Uniqueness is only recovered when an initial condition pins down the value of C.

Question 4

To find the antiderivative of xⁿ (for n ≠ −1), you increase the exponent by one and divide by the new exponent.

• T. True
• F. False

Answer: True

This is the power rule for antiderivatives: ∫xⁿ dx = x^(n+1)/(n+1) + C. It is the reverse of the derivative power rule. Differentiating x^(n+1)/(n+1) gives (n+1)·x^n/(n+1) = xⁿ, confirming the formula. The most common confusion is applying the derivative rule instead — multiplying by n and *decreasing* the exponent — which produces xⁿ⁻¹, not the antiderivative.

Question 5

Explain why we must always include '+C' when writing a general antiderivative.

Model answer: Because the derivative of any constant is zero, adding a constant to an antiderivative produces another valid antiderivative. The general antiderivative therefore represents a whole family of functions differing by a vertical shift. The +C captures this ambiguity; C can only be determined with additional information, such as a known point the function passes through.

Omitting +C implies the antiderivative is unique, which is false. This matters in applications: when solving a differential equation or modeling a physical system, the constant C encodes the initial condition (e.g., starting position or initial velocity). Without +C, you cannot apply those conditions and your solution will be wrong for all but one arbitrary case.