Chapter 16 — Exercises

46 problems on trigonometric integrals, trigonometric substitution, partial fraction decomposition, and numerical methods. Tiered ⭐ (warm-up) through ⭐⭐⭐⭐ (advanced). Work by hand first; verify with sympy or scipy second. Antiderivatives are unique only up to a constant — a sympy result that looks different may still be correct (§16.8).

Part A — Trigonometric Integrals (§16.2–16.3)

⭐ 16.1 $\displaystyle \int \sin^2 x \, dx$ (power reduction)

⭐ 16.2 $\displaystyle \int \cos^3 x \, dx$ (odd power — peel and substitute)

⭐ 16.3 $\displaystyle \int \tan^2 x \, dx$ (Hint: $\tan^2 x = \sec^2 x - 1$.)

⭐⭐ 16.4 $\displaystyle \int \sin^4 x \cos^3 x \, dx$

⭐⭐ 16.5 $\displaystyle \int \tan^3 x \sec^2 x \, dx$

⭐⭐ 16.6 $\displaystyle \int \sec^4 x \, dx$ (Hint: reserve one $\sec^2 x$, convert the rest.)

⭐⭐ 16.7 $\displaystyle \int \sin(3x) \cos(2x) \, dx$ (product-to-sum, §16.3)

⭐⭐ 16.8 $\displaystyle \int \sin^5 x \cos^2 x \, dx$

⭐⭐ 16.9 $\displaystyle \int \cos^4 x \, dx$ (double power-reduction)

⭐⭐⭐ 16.10 $\displaystyle \int_0^{\pi} \sin^4 x \, dx$ (definite — answer should be $\tfrac{3\pi}{8}$)

Part B — Trigonometric Substitution (§16.4)

⭐ 16.11 $\displaystyle \int \frac{dx}{\sqrt{9 - x^2}}$ (the $\arcsin$ form)

⭐⭐ 16.12 $\displaystyle \int \sqrt{4 - x^2} \, dx$

⭐⭐ 16.13 $\displaystyle \int \frac{dx}{x^2 \sqrt{x^2 + 1}}$

⭐⭐ 16.14 $\displaystyle \int \frac{dx}{x \sqrt{x^2 - 1}}$

⭐⭐ 16.15 $\displaystyle \int \frac{x^2}{\sqrt{1 - x^2}} \, dx$

⭐⭐⭐ 16.16 $\displaystyle \int \frac{dx}{(x^2 + 9)^2}$ (use $x = 3\tan\theta$; you will need $\int\cos^2\theta\,d\theta$)

⭐⭐ 16.17 $\displaystyle \int_0^1 \sqrt{1 - x^2} \, dx$ (should be $\pi/4$ — a quarter-circle area, cf. Worked Example 16.4.2)

⭐⭐⭐ 16.18 $\displaystyle \int \frac{dx}{\sqrt{4x^2 + 9}}$ (factor the $4$ out of the radical first, then $\sqrt{a^2+u^2}$ substitution)

Part C — Partial Fractions (§16.5)

⭐ 16.19 $\displaystyle \int \frac{dx}{x^2 - 4}$ (distinct linear factors, cover-up)

⭐⭐ 16.20 $\displaystyle \int \frac{x + 1}{x^2 + 5x + 6} \, dx$

⭐⭐ 16.21 $\displaystyle \int \frac{2x + 3}{(x - 1)(x + 2)} \, dx$

⭐⭐ 16.22 $\displaystyle \int \frac{dx}{x^3 - x}$ (factor $x^3 - x = x(x-1)(x+1)$)

⭐⭐⭐ 16.23 $\displaystyle \int \frac{x^2}{(x - 1)^3} \, dx$ (repeated linear factor — all three terms required)

⭐⭐⭐ 16.24 $\displaystyle \int \frac{x + 1}{x(x^2 + 1)} \, dx$ (linear + irreducible quadratic; expect a $\ln$ and an $\arctan$)

⭐⭐⭐ 16.25 $\displaystyle \int \frac{x^4}{x^2 - 1} \, dx$ (numerator degree $\ge$ denominator — long-divide first)

⭐⭐⭐⭐ 16.26 $\displaystyle \int \frac{dx}{(x^2 + 1)^2}$ (repeated irreducible quadratic — finish with trig substitution $x = \tan\theta$)

Part D — Numerical Integration (§16.6)

⭐⭐ 16.27 Use the trapezoidal rule with $n = 4$ to estimate $\displaystyle \int_0^1 e^{x^2} \, dx$. Compare with scipy.integrate.quad (reference value $\approx 1.46265$).

⭐⭐ 16.28 Use Simpson's rule with $n = 4$ on $\displaystyle \int_0^1 e^{x^2} \, dx$. Confirm it beats your $n=4$ trapezoidal estimate from 16.27.

⭐⭐ 16.29 Estimate $\displaystyle \int_0^1 \sin(x^2) \, dx$ with $n = 100$ using Simpson's rule (reference $\approx 0.310268$).

⭐⭐⭐ 16.30 Estimate $\displaystyle \int_0^{1} \frac{\sin x}{x} \, dx$ using both rules with $n=10$. (The integrand is removable at $x = 0$ if you set $f(0) = 1$; the limit $\lim_{x\to0}\frac{\sin x}{x}=1$ comes from Chapter 3.)

⭐⭐⭐ 16.31 Implement Simpson's rule in Python without scipy's built-in, and apply it to $\int_0^\pi \sin x \, dx$ (exact answer $2$). Report the error at $n = 10$ and $n = 100$; confirm it shrinks by roughly $10^4$.

⭐⭐⭐ 16.32 Use the trapezoidal error bound $|I - T_n| \le \frac{(b-a)^3}{12n^2}\max|f''|$ to determine how large $n$ must be to guarantee $|I - T_n| < 10^{-4}$ for $\int_0^1 e^{-x^2}\,dx$. (You will need $\max_{[0,1]}|f''|$; note $f''(x) = (4x^2-2)e^{-x^2}$.)

Part E — Mixed Practice (⭐⭐⭐⭐)

⭐⭐⭐ 16.33 $\displaystyle \int \frac{x^2 + 1}{x^4 - 1} \, dx$ (simplify the fraction before decomposing)

⭐⭐⭐ 16.34 $\displaystyle \int \frac{\sec^2 x}{1 + \tan x} \, dx$ (spot the inner derivative)

⭐⭐⭐⭐ 16.35 $\displaystyle \int \csc x \, dx = \int \frac{dx}{\sin x}$ (multiply top and bottom by $\sin x$, then $u = \cos x$, then partial fractions)

⭐⭐⭐ 16.36 $\displaystyle \int \frac{dx}{1 + e^x}$ (substitute $u = e^x$ or rewrite $\frac{1}{1+e^x} = 1 - \frac{e^x}{1+e^x}$)

⭐⭐⭐⭐ 16.37 $\displaystyle \int \frac{dx}{\sqrt{x^2 + 4x + 5}}$ (complete the square to $(x+2)^2 + 1$, then $\sqrt{a^2+u^2}$ substitution)

⭐⭐⭐ 16.38 $\displaystyle \int x \arctan x \, dx$ (integration by parts from Chapter 15, then a partial-fraction-style split of $\frac{x^2}{x^2+1}$)

Part F — Applications (⭐⭐⭐ unless marked)

⭐⭐⭐ 16.39 — Standard-normal probability (statistics, data science). Use Simpson's rule with $n = 100$ on $\displaystyle\int_0^1 \frac{1}{\sqrt{2\pi}} e^{-x^2/2}\,dx$. This equals $P(0 < Z < 1) \approx 0.3413$ for a standard normal — the "Area Under the Normal Curve" anchor from Chapter 13. The integrand has no elementary antiderivative, so numerical methods are mandatory.

⭐⭐⭐ 16.40 — Arc length of a parabola (physics, engineering). Compute $\displaystyle\int_0^1 \sqrt{1 + 4x^2}\,dx$ by trigonometric substitution ($2x = \tan\theta$). This is the arc length of $y = x^2$ from $x=0$ to $x=1$; the closed form is $\tfrac14\sinh^{-1}2 + \tfrac{\sqrt5}{2} \approx 1.4789$. Confirm numerically with quad.

⭐⭐⭐⭐ 16.41 — Volume of a wine barrel (engineering). A barrel has circular cross-section of radius $r(z) = R - c z^2$ for $|z| \le h$. Its volume is $V = \displaystyle\int_{-h}^{h} \pi\, r(z)^2\,dz$. Expand the square and integrate term by term to show $V = 2\pi\!\left(R^2 h - \tfrac{2}{3}R c h^3 + \tfrac{1}{5}c^2 h^5\right)$. Then evaluate for $R = 0.4\,\text{m}$, $c = 0.6\,\text{m}^{-1}$, $h = 0.5\,\text{m}$.

⭐⭐⭐ 16.42 — Marginal cost to total cost (economics). A factory's marginal cost (dollars per unit) is measured at production levels $q = 0, 20, 40, 60, 80, 100$ units as $C'(q) = 50,\,42,\,38,\,37,\,40,\,46$. The added cost of ramping from $0$ to $100$ units is $\int_0^{100} C'(q)\,dq$. Estimate it with Simpson's rule (note $n = 5$ subintervals is odd — use the composite trapezoidal rule instead, or Simpson on the first four panels plus one trapezoid). Explain why you cannot apply plain Simpson's rule directly.

Part G — Reflective & Exploratory (⭐⭐⭐⭐)

⭐⭐⭐ 16.43 Compute $\displaystyle\int \frac{dx}{x(x+1)}$ three ways: (a) partial fractions, (b) the substitution $u = x+1$, and (c) the algebraic identity $\frac{1}{x(x+1)} = \frac1x - \frac{1}{x+1}$. Verify all three agree up to a constant.

⭐⭐⭐⭐ 16.44 Find an integral solvable by both trigonometric substitution and a clever $u$-substitution — for instance $\int \frac{x\,dx}{\sqrt{1-x^2}}$. Solve it both ways and reconcile the answers.

⭐⭐⭐ 16.45 Name three integrals with no elementary antiderivative besides $\int e^{-x^2}\,dx$. For each, identify the "special function" it defines (e.g. the error function $\operatorname{erf}$, the sine integral $\operatorname{Si}$, the Fresnel integral). Use sympy.integrate to see what it returns.

⭐⭐⭐⭐ 16.46 Estimate $\displaystyle\int_0^{3} e^{-x^2}\,dx$ to 8 correct decimals using progressively finer Simpson approximations ($n = 10, 20, 40, \dots$). How many subintervals does 8-digit accuracy require? Compare to the trapezoidal rule's requirement, and connect your finding to the $1/n^4$ vs. $1/n^2$ error bounds of §16.6.

Difficulty Distribution

Total: 46 exercises, ~8–12 hours. Applications (Part F) span statistics/data science, physics/engineering, and economics; the ⭐⭐⭐⭐ tier appears in Parts C, E, F, and G.

Solutions

Selected fully worked solutions appear in appendices/answers-to-selected.md.