Antiderivatives & integrals
Applications: optimization and related rates sharpen the “constraint + differentiate” habit for substitution.
Learning goals
- Reverse differentiation: write general antiderivatives with $+C$.
- Integrate sums, scaled terms, and basic substitutions (“$u$-substitution”).
- Interpret $\int_a^b f(x)\,dx$ as net signed area (before FTC machinery in depth).
- Recognize when substitution needs a compensating factor from the chain rule.
Concept sheet
Antiderivative: $F$ is an antiderivative of $f$ on an interval if $F'(x)=f(x)$ there. All antiderivatives differ by a constant: $\int f(x)\,dx = F(x)+C$.
Linearity: $\int (af+bg)\,dx = a\int f\,dx + b\int g\,dx$ (on an interval where both are integrable in your sense).
If $u = g(x)$, then $du = g'(x)\,dx$. Rewrite the integrand as “$f(u)\,du$” up to a constant factor you can balance.
Example: $\displaystyle \int 2x\cos(x^2)\,dx$. Let $u=x^2$, $du=2x\,dx$, so integral is $\int \cos u\,du = \sin u + C = \sin(x^2)+C$.
Worked patterns
Pattern A — guess-and-check with chain correction
$\int e^{3x}\,dx$: guess $e^{3x}$; derivative picks up factor $3$, so $\frac{1}{3}e^{3x}+C$.
Pattern B — trig antiderivatives
$\int \sin x\,dx = -\cos x + C$, $\int \cos x\,dx = \sin x + C$.
Pattern C — rational setup (partial fractions if your course includes)
Otherwise defer to algebra simplification first (long division on improper rational degree).
Practice
Original drills — use Check when available, then Show solution. A quick celebration plays on correct auto-checks (respects reduced motion).
Homework / practice
The syllabus continues with §5.1–§5.5 (areas, definite integral, FTC, substitution). There are
no matching WebAssign PDFs in materials/ yet — add them when you have
exports and we can link rows here and on FTC & area.
| Section | Link | Focus |
|---|---|---|
| 4.9 — Antiderivatives | Open PDF | Indefinite integrals, $+C$, building toward FTC |
Mistakes & checks
- Forgetting $+C$ on indefinite integrals.
- Substitution: must account for $du$; if a factor is off by constant, adjust; if structural mismatch, try another $u$.
- Domain issues: $\int \frac{1}{x}\,dx = \ln|x|+C$ on $(0,\infty)$ or $(-\infty,0)$ separately — many courses use $\ln|x|+C$ as shorthand.
Slide appendix (optional)
Add a short exported PDF here if you want slide backup for substitution drills.
Quiz mode
Homework PDF quiz: each card links to the same WebAssign export filenames as the semester map (materials/). A session shows 20 shuffled cards from 45 PDF-linked slots; exact problem wording stays in the PDF unless you merge transcripts per quiz/transcripts/README.md. Regenerate: python3 scripts/gen_hw_quiz.py.