A student computes ∫6x² dx = 2x³ and considers the problem finished. What is wrong with this answer?
ANothing — 2x³ is the correct and complete antiderivative of 6x²
BThe constant of integration +C is missing; without it, only one antiderivative is named instead of the entire family
CThe dx should appear in the answer alongside the result
DThe integral sign should be retained in the answer to show the operation is ongoing
Differentiation destroys any constant: (F(x) + C)' = F'(x) for any constant C. This means if F'(x) = f(x), then (F(x) + C)' = f(x) as well, for every value of C. The family 2x³ + C represents ALL antiderivatives of 6x². Writing just 2x³ implicitly claims there is only one antiderivative, which is false. Omitting +C is an incomplete answer, not merely a notational preference.
Question 2 Multiple Choice
What is the key difference between the indefinite integral ∫f(x) dx and the definite integral ∫ₐᵇ f(x) dx?
ABoth produce functions of x, but the definite integral restricts the domain to [a, b]
BThe indefinite integral produces a family of functions; the definite integral produces a specific number
CThey produce the same result — the definite integral just adds evaluation at the bounds
DThe indefinite integral includes +C while the definite integral does not, but otherwise they are equivalent objects
The indefinite integral ∫f(x) dx = F(x) + C is a function (or family of functions). The definite integral ∫ₐᵇ f(x) dx = F(b) − F(a) is a number — the net signed area under f between a and b. Option D is the most tempting wrong answer: while it's true that the +C cancels in F(b) − F(a), the objects themselves are fundamentally different in kind (function vs. number), not merely differing by a constant.
Question 3 True / False
The dx in ∫f(x) dx is optional notation that can be omitted without affecting the mathematical meaning.
TTrue
FFalse
Answer: False
The dx identifies the variable of integration — essential when the integrand involves multiple variables, and critical when performing substitution. In u-substitution, the differential itself transforms: if u = g(x), then du = g'(x) dx, and the dx in the original integral becomes part of the substitution. Omitting dx makes this transformation impossible to execute. It is a structural part of the notation, not decorative.
Question 4 True / False
The indefinite integral of a function f(x) represents the family of all functions that differentiate to f(x).
TTrue
FFalse
Answer: True
This is the definition. If F'(x) = f(x), then every function of the form F(x) + C (for any constant C) also differentiates to f(x). The indefinite integral names this entire family. The +C encodes the fact that constants vanish under differentiation, so the antiderivative is not unique — it is determined only up to an additive constant.
Question 5 Short Answer
How do you verify that an indefinite integral is correct, and why does this method always work?
Think about your answer, then reveal below.
Model answer: Differentiate the result and check whether you recover the original integrand. If ∫f(x) dx = F(x) + C, compute d/dx[F(x) + C] and confirm it equals f(x). This always works because integration and differentiation are inverse operations — the derivative of an antiderivative of f must return f. The +C disappears in differentiation (the derivative of any constant is zero), so it doesn't affect the check.
This verification strategy is available for every indefinite integral and should become automatic. If d/dx[your answer] ≠ f(x), the integral is wrong. The method works because the Fundamental Theorem of Calculus guarantees that differentiation undoes integration: d/dx ∫f(x) dx = f(x). The check exploits this inverse relationship directly.