Chapter 16 — Self-Assessment Quiz

10 questions, ~20 minutes. Aim for 8/10. Each answer cites the section to revisit if you miss it.

1. For $\displaystyle\int \sin^2 x \, dx$, the most efficient technique is: - A) $u$-substitution with $u = \sin x$ - B) Power-reduction identity $\sin^2 x = \tfrac{1 - \cos 2x}{2}$ - C) Integration by parts - D) Trigonometric substitution

Answer**B.** Both powers are even (here $m=2$, $n=0$), so no factor can be peeled off as a clean differential — power reduction is the only route. The result is $\tfrac{x}{2} - \tfrac{\sin 2x}{4} + C$. *Reference: §16.2, Case 3.*

2. For $\displaystyle\int \sqrt{4 - x^2} \, dx$, the appropriate substitution is: - A) $x = 2\sin\theta$ - B) $x = 2\tan\theta$ - C) $x = 2\sec\theta$ - D) None — use partial fractions

Answer**A.** This is the $\sqrt{a^2 - x^2}$ pattern with $a = 2$. Setting $x = 2\sin\theta$ turns the radical into $\sqrt{4 - 4\sin^2\theta} = 2\cos\theta$, eliminating it. *Reference: §16.4 (substitution table) and Worked Example 16.4.2.*

3. Partial fraction decomposition applies to: - A) Any product of trig functions - B) Rational functions $p(x)/q(x)$ with $p,q$ polynomials - C) Exponential integrands - D) Integrands containing a radical

Answer**B.** Partial fractions split a ratio of polynomials into simpler fractions you can integrate term by term. If $\deg p \ge \deg q$, divide first. *Reference: §16.5.*

4. The decomposition $\dfrac{1}{(x-1)(x+1)} = \dfrac{A}{x - 1} + \dfrac{B}{x + 1}$ gives: - A) $A = \tfrac12,\ B = \tfrac12$ - B) $A = \tfrac12,\ B = -\tfrac12$ - C) $A = 1,\ B = -1$ - D) $A = -1,\ B = 1$

Answer**B.** Clear denominators: $1 = A(x+1) + B(x-1)$. Cover-up at $x = 1$ gives $1 = 2A \Rightarrow A = \tfrac12$; at $x = -1$ gives $1 = -2B \Rightarrow B = -\tfrac12$. *Reference: §16.5, Worked Example 16.5.1.*

5. The trapezoidal-rule error scales like: - A) $1/n$ - B) $1/n^2$ - C) $1/n^3$ - D) $1/n^4$

Answer**B.** $|I - T_n| \le \frac{(b-a)^3}{12n^2}\max|f''|$, so doubling $n$ quarters the error. Simpson's rule, by contrast, scales like $1/n^4$. *Reference: §16.6.*

6. Simpson's rule requires: - A) Any $n$ - B) $n$ even - C) $n$ odd - D) $n$ a power of $2$

Answer**B.** Simpson's rule fits a parabola through each consecutive *triple* of nodes, so the subintervals must pair up — $n$ must be even. The weight pattern is $1,4,2,4,\dots,4,1$. *Reference: §16.6.*

7. $\displaystyle\int \frac{dx}{x^2 + 1} = $ - A) $\arctan x + C$ - B) $\arcsin x + C$ - C) $\tfrac12\ln(x^2 + 1) + C$ - D) $\arctan(x^2) + C$

Answer**A.** A standard antiderivative; it also falls out of the trig substitution $x = \tan\theta$, which converts the integrand to $\int d\theta$. The constant left over a generic irreducible quadratic always integrates to an $\arctan$. *Reference: §16.4 and §16.5.*

8. $\displaystyle\int \tan^2 x \, dx = $ - A) $\tan x + C$ - B) $\sec^2 x + C$ - C) $\tan x - x + C$ - D) $\sec x \tan x + C$

Answer**C.** Rewrite $\tan^2 x = \sec^2 x - 1$, then integrate: $\int(\sec^2 x - 1)\,dx = \tan x - x + C$. *Reference: §16.2, Case 4.*

9. Which integral has no elementary antiderivative? - A) $\int x^2\,dx$ - B) $\int e^{x}\,dx$ - C) $\int e^{-x^2}\,dx$ - D) $\int \sin x\,dx$

Answer**C.** $\int e^{-x^2}\,dx$ is the classic example (Liouville's theorem); the related error function $\operatorname{erf}(x)$ is *defined* by this integral, which is exactly why numerical methods exist. *Reference: §16.6 and §16.7.*

10. In a partial-fraction decomposition, an irreducible quadratic factor $x^2 + bx + c$ contributes a term of the form: - A) $\dfrac{A}{x^2 + bx + c}$ - B) $\dfrac{Bx + C}{x^2 + bx + c}$ - C) $\dfrac{A}{x + r}$ - D) $\dfrac{B}{x - r} + \dfrac{C}{x + r}$

Answer**B.** An irreducible quadratic (no real roots, $b^2 - 4c < 0$) gets a *linear* numerator $Bx + C$. Writing only a constant on top is the most common partial-fractions error. *Reference: §16.5, Worked Example 16.5.3.*

Scoring

  • 9–10: Excellent — you own the full integration toolkit. Move on to Chapter 17.
  • 7–8: Solid. Re-skim the section any missed question points to.
  • 5–6: Re-read §16.5 (partial fractions) and §16.4 (trig substitution); these trip up the most students. Redo every Worked Example.
  • Below 5: Slow down and reread the whole chapter, working each Worked Example by hand before reading its solution.