Chapter 16 — Key Takeaways
Three analytic techniques plus one numerical fallback complete your integration toolkit. The skill is no longer brute force — it is recognizing which disguise to strip away, and knowing when no disguise exists.
1. Trigonometric Integrals (§16.2–16.3)
The strategy is always the same: use an identity to free one factor as the differential, then substitute. Choose by the parity of the powers.
| Form | Trigger | Move |
|---|---|---|
| $\int \sin^m x\cos^n x\,dx$, $m$ odd | spare $\sin x$ | peel one $\sin x$ for $du$, convert rest via $\sin^2 = 1-\cos^2$, let $u = \cos x$ |
| $\int \sin^m x\cos^n x\,dx$, $n$ odd | spare $\cos x$ | peel one $\cos x$ for $du$, convert rest via $\cos^2 = 1-\sin^2$, let $u = \sin x$ |
| $\int \sin^m x\cos^n x\,dx$, both even | no spare factor | power reduction $\sin^2 = \tfrac{1-\cos 2x}{2}$, $\cos^2 = \tfrac{1+\cos 2x}{2}$ (repeat as needed) |
| $\int \tan^m x\sec^n x\,dx$, $n$ even | spare $\sec^2 x$ | peel $\sec^2 x$ for $du$, convert via $\sec^2 = 1+\tan^2$, let $u = \tan x$ |
| $\int \tan^m x\sec^n x\,dx$, $m$ odd | spare $\sec x\tan x$ | peel $\sec x\tan x$ for $du$, convert via $\tan^2 = \sec^2 - 1$, let $u = \sec x$ |
| $\int \sin(mx)\cos(nx)\,dx$ (§16.3) | different arguments | product-to-sum identity, then integrate each term |
Mnemonic: odd power → peel and substitute; both even → half-angle.
2. Trigonometric Substitution (§16.4)
When a radical $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$ appears, replace $x$ by a trig function so a Pythagorean identity collapses the radical.
| Radical | Substitution | $dx$ | Radical becomes |
|---|---|---|---|
| $\sqrt{a^2 - x^2}$ | $x = a\sin\theta$ | $a\cos\theta\,d\theta$ | $a\cos\theta$ |
| $\sqrt{a^2 + x^2}$ | $x = a\tan\theta$ | $a\sec^2\theta\,d\theta$ | $a\sec\theta$ |
| $\sqrt{x^2 - a^2}$ | $x = a\sec\theta$ | $a\sec\theta\tan\theta\,d\theta$ | $a\tan\theta$ |
The right-triangle picture (so you never memorize back-substitution). Each row is a right triangle:
- $\sqrt{a^2 - x^2}$: hypotenuse $a$, one leg $x$, other leg $\sqrt{a^2-x^2}$; $\sin\theta = x/a$.
- $\sqrt{a^2 + x^2}$: legs $x$ and $a$, hypotenuse $\sqrt{a^2+x^2}$; $\tan\theta = x/a$.
- $\sqrt{x^2 - a^2}$: hypotenuse $x$, one leg $a$, other leg $\sqrt{x^2-a^2}$; $\sec\theta = x/a$.
After integrating in $\theta$, read every trig function of $\theta$ off the triangle to convert back to $x$ — or, for a definite integral, change the limits and skip back-substitution entirely.
3. Partial Fraction Decomposition (§16.5)
Every rational function $p(x)/q(x)$ splits into pieces you can integrate by inspection.
- Divide first if $\deg p \ge \deg q$ (long division) — decomposition only applies to proper fractions.
- Factor $q(x)$ into linear factors $(x-r)$ and irreducible quadratics $(x^2+bx+c)$ with $b^2 - 4c < 0$.
- Write the template — the four cases:
| Factor in $q(x)$ | Contributes to the template |
|---|---|
| Distinct linear $(x-r)$ | $\dfrac{A}{x-r}$ |
| Repeated linear $(x-r)^k$ | $\dfrac{A_1}{x-r} + \dfrac{A_2}{(x-r)^2} + \cdots + \dfrac{A_k}{(x-r)^k}$ (all $k$ terms) |
| Distinct irreducible quadratic $(x^2+bx+c)$ | $\dfrac{Bx+C}{x^2+bx+c}$ (linear numerator) |
| Repeated irreducible quadratic $(x^2+bx+c)^k$ | $\dfrac{B_1x+C_1}{x^2+bx+c} + \cdots + \dfrac{B_kx+C_k}{(x^2+bx+c)^k}$ |
- Solve for coefficients: cover-up (substitute roots) is fastest for linear factors; coefficient matching is best for quadratics.
- Integrate each piece: linear factors give $\ln$; the $x$ part of a quadratic gives $\ln$ and the constant part gives $\arctan$.
4. Numerical Integration (§16.6)
When no elementary antiderivative exists, or the integrand is data, approximate the definite integral. Partition $[a,b]$ into $n$ subintervals of width $h = (b-a)/n$.
| Rule | Formula | Weights | Error bound | Requires |
|---|---|---|---|---|
| Trapezoidal | $T_n = \dfrac{h}{2}\big[f_0 + 2f_1 + \cdots + 2f_{n-1} + f_n\big]$ | $1,2,2,\dots,2,1$ | $\dfrac{(b-a)^3}{12n^2}\max\lvert f''\rvert$ | any $n$ |
| Simpson's | $S_n = \dfrac{h}{3}\big[f_0 + 4f_1 + 2f_2 + 4f_3 + \cdots + 4f_{n-1} + f_n\big]$ | $1,4,2,4,\dots,4,1$ | $\dfrac{(b-a)^5}{180n^4}\max\lvert f^{(4)}\rvert$ | $n$ even |
- Trapezoidal error $\sim 1/n^2$: double $n$, error drops by $4$.
- Simpson error $\sim 1/n^4$: double $n$, error drops by $16$. Exact for any cubic.
- The error bound certifies precision without knowing the exact answer — its real value when no closed form exists.
scipy.integrate.quad(adaptive quadrature, built on QUADPACK) is the production default; it samples densely where the integrand varies and returns a rigorous error estimate.
When to go numerical
- No elementary antiderivative — $e^{-x^2}$, $\sin(x^2)$, $\tfrac{\sin x}{x}$, the elliptic (ellipse-perimeter) integral.
- The integrand is data, not a formula — sensor readings, a column in a spreadsheet. Trapezoid/Simpson on the sample points is the only option.
- A closed form exists but is hideous — sometimes a number is all you need.
Boundary warning: trapezoid and Simpson assume a finite interval with a bounded integrand. Infinite limits or infinite integrands are improper integrals — the subject of Chapter 17. Do not point these rules at such an integral first.
5. Standard Results to Memorize
$$\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\frac{x}{a} + C, \qquad \int \frac{dx}{a^2 + x^2} = \frac1a\arctan\frac{x}{a} + C,$$ $$\int \frac{dx}{\sqrt{x^2 + a^2}} = \ln\big|x + \sqrt{x^2 + a^2}\big| + C, \qquad \int \frac{dx}{x-r} = \ln|x-r| + C,$$ $$\int \cos^2 x\,dx = \frac{x}{2} + \frac{\sin 2x}{4} + C, \qquad \int \sin^2 x\,dx = \frac{x}{2} - \frac{\sin 2x}{4} + C.$$
6. The Master Decision Tree (§16.7)
| Integrand structure | Technique |
|---|---|
| Basic form in the table | Direct lookup (Ch. 12) |
| Composite × inner derivative | $u$-substitution (Ch. 15) |
| Product of unlike function types | Integration by parts, LIATE (Ch. 15) |
| $\sin^m x\cos^n x$ or $\tan^m x\sec^n x$ | §16.2–16.3 |
| Radical $\sqrt{a^2 \pm x^2}$ or $\sqrt{x^2 - a^2}$ | Trig substitution (§16.4) |
| Rational function $p(x)/q(x)$ | Partial fractions (§16.5) |
| No elementary antiderivative, or data | Numerical (§16.6) |
7. Common Errors to Avoid
- Trig integrals — wrong tool for the parity. Trying $u = \cos x$ on $\int\sin^4 x\,dx$ fails: there is no spare $\sin x$. Both-even powers need power reduction, never peel-and-substitute.
- Trig substitution — dropping the absolute value. $\sqrt{a^2\sin^2\theta} = a|\cos\theta|$; you may drop the bars only after checking your $\theta$-range keeps the function nonnegative (always an issue for $x = a\sec\theta$).
- Trig substitution — forgetting to back-substitute. "$\tan\theta - \theta + C$" is not an answer to a problem posed in $x$. Rebuild every trig function via the triangle, or change the limits.
- Partial fractions — too small a template for a repeated factor. $(x-r)^3$ demands all three terms $\frac{A_1}{x-r} + \frac{A_2}{(x-r)^2} + \frac{A_3}{(x-r)^3}$, not just the last.
- Partial fractions — constant numerator on an irreducible quadratic. Write $\frac{Bx+C}{x^2+bx+c}$, never $\frac{A}{x^2+bx+c}$.
- Partial fractions — forgetting to divide first. Decomposition applies only to proper fractions ($\deg p < \deg q$).
- Numerical — using odd $n$ with Simpson's rule. Simpson requires $n$ even; odd $n$ misapplies the $1,4,2,\dots$ weights.
What's Next
- Chapter 17 — Improper integrals. Lifts the "finite interval, bounded integrand" restriction, with convergence tests deciding when $\int_1^\infty \tfrac{dx}{x^2}$ or $\int_0^1 \tfrac{dx}{\sqrt{x}}$ even has a value.
- Chapter 18 — Applications of integration. Turns the whole toolkit loose on area, volume, arc length, surface area, and work.
- Chapter 19 — Differential equations. Closes Part III; the partial-fraction technique returns to solve the logistic model.
Reflection
You now hold every major integration technique. From here on, "evaluate the integral" is a step inside a larger modeling problem, never the whole problem. If you can work every exercise in Chapters 15 and 16 fluently, you have the integration skill to carry through the rest of this book, any standard physics or engineering text, and most applied-mathematics papers in your field.