Chapter 12 — Quiz

10 questions covering antiderivatives, the indefinite integral, linearity, initial value problems, kinematics, and the boundary of elementary integration. Each answer cites the section it comes from. Aim to verify every integral by differentiating it back (Section 12.13).

1. A function $F$ is an antiderivative of $f$ when:

  • A) $F'' = f$ B) $F' = f$ C) $F = f'$ D) $F = -f$
Answer**B) $F' = f$.** Antidifferentiation is the inverse of differentiation: $F$ is an antiderivative of $f$ exactly when differentiating $F$ returns $f$ (Section 12.1).

2. $\displaystyle\int x^3\,dx = $

  • A) $3x^2 + C$ B) $x^4 + C$ C) $\dfrac{x^4}{4} + C$ D) $4x^4 + C$
Answer**C) $\dfrac{x^4}{4} + C$.** Power rule (Section 12.5): raise the exponent to $4$, divide by $4$. Check: $\left(\tfrac{x^4}{4}\right)' = x^3$. ✓

3. $\displaystyle\int \frac{1}{x}\,dx = $

  • A) $\ln x + C$ B) $\ln\lvert x\rvert + C$ C) $-\dfrac{1}{x^2} + C$ D) $\dfrac{x^0}{0} + C$
Answer**B) $\ln\lvert x\rvert + C$.** This is the $n=-1$ exception the power rule cannot touch (Section 12.6). The absolute value extends the antiderivative to negative $x$. Option D is the nonsense you get from forcing the power rule.

4. $\displaystyle\int \sin x\,dx = $

  • A) $\cos x + C$ B) $-\cos x + C$ C) $\sin x + C$ D) $-\sin x + C$
Answer**B) $-\cos x + C$.** The minus sign lives with $\sin$ (Section 12.4). Check: $(-\cos x)' = \sin x$. ✓

5. Why does every indefinite integral carry a "$+C$"?

  • A) Because integration is hard
  • B) Because differentiation forgets constants, so antidifferentiation cannot recover them — antiderivatives form a family differing by a constant
  • C) Because the integral sign demands it
  • D) Because $C$ stands for the curve
Answer**B.** The derivative of any constant is zero, so $F(x)$ and $F(x)+C$ have the same derivative. By the "antiderivatives differ by a constant" theorem (a consequence of the Mean Value Theorem), the whole family is captured by one antiderivative plus an arbitrary $C$ (Section 12.2).

6. Linearity lets you rewrite $\displaystyle\int (3f(x) + 2g(x))\,dx$ as:

  • A) $3\displaystyle\int f\,dx + 2\displaystyle\int g\,dx$
  • B) $5\displaystyle\int (f+g)\,dx$
  • C) $\left(\displaystyle\int f\,dx\right)\left(\displaystyle\int g\,dx\right)$
  • D) It cannot be split
Answer**A.** Integration is linear: sums split and constant factors slide out front (Section 12.7). But linearity covers *only* sums and constant multiples — there is no product rule for integrals, ruling out option C.

7. $\displaystyle\int \cos(3x)\,dx = $

  • A) $\sin(3x) + C$ B) $3\sin(3x) + C$ C) $\dfrac{1}{3}\sin(3x) + C$ D) $-\dfrac{1}{3}\sin(3x) + C$
Answer**C) $\dfrac{1}{3}\sin(3x) + C$.** Reading the chain rule backward for the linear interior $3x$: integrate to $\sin(3x)$, then divide by the inner slope $3$ (Section 12.7½). Check: $\left(\tfrac13\sin 3x\right)' = \tfrac13 \cdot 3\cos 3x = \cos 3x$. ✓

8. Solve the IVP $f'(x) = 2x$ with $f(1) = 5$.

  • A) $f(x) = x^2 + 5$ B) $f(x) = x^2 + 4$ C) $f(x) = 2x^2 + 3$ D) $f(x) = 2x + 3$
Answer**B) $f(x) = x^2 + 4$.** Antidifferentiate: $f(x) = x^2 + C$. Apply $f(1) = 1 + C = 5 \Rightarrow C = 4$ (Section 12.8). The initial condition selects one parabola from the family $x^2 + C$.

9. A particle has acceleration $a(t)$. To recover its position $s(t)$, you must antidifferentiate twice, and you need:

  • A) one initial condition (position only)
  • B) two initial conditions (an initial velocity and an initial position)
  • C) no initial conditions
  • D) the final position only
Answer**B.** Each antidifferentiation introduces one constant: integrating $a \to v$ needs $v(0)$, and integrating $v \to s$ needs $s(0)$. Two integrations, two conditions (Section 12.10).

10. Which statement about $\displaystyle\int e^{-x^2}\,dx$ is correct?

  • A) It equals $-2x\,e^{-x^2} + C$
  • B) It has no antiderivative at all
  • C) An antiderivative exists, but it is non-elementary — it defines the error function $\mathrm{erf}$
  • D) It equals $\dfrac{e^{-x^2}}{-2x} + C$
Answer**C.** By Liouville's theorem the antiderivative cannot be written with finitely many elementary functions, but it still *exists* (every continuous function has one — proven in Chapter 14); mathematicians name it $\mathrm{erf}$ (Section 12.9). Options A and D are wrong because you cannot divide by the non-constant slope $-2x$.

Scoring Guide

  • 9–10 correct — Excellent. You command the basic table, linearity, the chain-rule-backward trick, and IVPs. You are ready for the definite integral in Chapter 13.
  • 7–8 correct — Solid. Review whichever section the missed questions cite, then re-drill the basic antiderivative table (Section 12.4).
  • 5–6 correct — Developing. Re-read Sections 12.4–12.8 and redo Parts A, B, and F of the exercises until the table is automatic.
  • Below 5 — Revisit the chapter. Focus on the table (12.4), the power rule (12.5), and the $n=-1$ exception (12.6); these are the foundation everything else rests on.