Improper Integrals (4.8)
| Type | Definition | Key Rule |
|---|
| I (infinite limit) | $\int_a^\infty f,dx = \lim_{b\to\infty}\int_a^b f,dx$ | Replace $\infty$ with $b$, take limit |
| II (discontinuity) | $\int_a^b f,dx = \lim_{c\to a^+}\int_c^b f,dx$ | Replace bad bound with $c$, take limit |
| Interior at $d$ | MUST split → $\int_a^d + \int_d^b$ | One diverges → whole thing DIVERGES |
$p$-Test & Convergence
| Function | At $\infty$ ($\int_1^\infty$) | At $0$ ($\int_0^1$) |
|---|
| $1/x^2$ $(p{=}2)$ | Converges | Diverges |
| $1/x$ $(p{=}1)$ | Diverges | Diverges |
| $1/\sqrt{x}$ $(p{=}\frac{1}{2})$ | Diverges | Converges |
| $\ln(x)$ | — | Always converges |
| $\sin(x)/x$ | — | Always converges ($\to 1$) |
Rule: At $\infty$: $p > 1$ converges. At $0$: $p < 1$ converges. Directions flip!
Traps:
- $\int_{-1}^1 \frac{1}{x^2}dx$ — interior blow-up at $0$ → divergent
- $\sin(x)/x \to 1$ → bounded → convergent (no computation needed)
- Write “DIVERGES” not “$= \infty$”
- Odd/Even: $f$ odd on $[-a,a]$: $\int = 0$ · $f$ even: $\int = 2\int_0^a$. Must verify both halves converge first.
Volume (4.9–4.10)
| Method | Formula | When |
|---|
| Disk | $\pi\int_a^b [f(x)]^2;dx$ | Solid, no hole, $\perp$ to axis |
| Washer | $\pi\int_a^b [R(x)^2 - r(x)^2];dx$ | Two curves, hole in middle |
| Shell | $2\pi\int_a^b (\text{radius})(\text{height});dx$ | Parallel to axis |
| Cross-section | $\int_a^b A(x);dx$ | Non-revolution |
| Cross-section ($s$ = side) | Area |
|---|
| Square | $s^2$ |
| Equil. triangle | $\frac{\sqrt{3}}{4}s^2$ |
| Isosc. right triangle (leg $s$) | $\frac{1}{2}s^2$ |
| Semicircle (dia $s$) | $\frac{\pi}{8}s^2$ |
| Circle (dia $s$) | $\frac{\pi}{4}s^2$ |
| Axis | Method | Shifted | Method | Shift radius |
|---|
| $x$-axis | Disk/Washer | $y=k$ | Disk/Washer | $f(x)-k$ or $k-f(x)$ |
| $y$-axis | Shell | $x=k$ | Shell | $\lvert x-k\rvert$ |
Standard Antiderivatives/Integral (+ C)
| $f(x)$ | $\int f\,dx$ | $f(x)$ | $\int f\,dx$ | $f(x)$ | $\int f\,dx$ |
|---|
| $x^n$ | $\frac{x^{n+1}}{n+1}$ | $\frac{1}{x}$ | $\ln|x|$ | $e^x$ | $e^x$ |
| $e^{-x}$ | $-e^{-x}$ | $\sin x$ | $-\cos x$ | $\cos x$ | $+\sin x$ |
| $\sec^2 x$ | $\tan x$ | $\csc^2 x$ | $-\cot x$ | $\sec x\tan x$ | $\sec x$ |
| $\csc x\cot x$ | $-\csc x$ | $\tan x$ | $\ln|\sec x|$ | $\cot x$ | $\ln|\sin x|$ |
| $\frac{1}{1+x^2}$ | $\tan^{-1} x$ | $\frac{1}{\sqrt{1-x^2}}$ | $\sin^{-1} x$ | $\ln(t)$ | $t\ln t - t$ |
| $\frac{1}{a^2+x^2}$ | $\frac{1}{a}\tan^{-1}\frac{x}{a}$ | $\frac{1}{\sqrt{a^2-x^2}}$ | $\sin^{-1}\frac{x}{a}$ | Trap: $\div$ NEW exp $(n{+}1)$, not old! |
Differentiation ↔ Integration
| Rule | Differentiation | Reverse (Integration) |
|---|
| Product ↔ IBP | $(uv)’ = u’v + uv'$ | $\int u,dv = uv - \int v,du$ |
| Chain ↔ u-sub | $[f(g(x))]’ = f’(g(x)) \cdot g’(x)$ | $\int f(g(x)) \cdot g’(x),dx = \int f(u),du$ |
| Quotient | $(\frac{u}{v})’ = \frac{u’v - uv’}{v^2}$ | rewrite as product: $u \cdot v^{-1}$ |
IBP: Choose u (LIATE) → find du, v → apply $uv - \int v,du$
U-sub: Spot inner $u=g(x)$ → compute $du=g’(x),dx$ → replace all $x$ → integrate → sub back
u-sub bounds: set $u=g(x)$, then $x=a \to u=g(a)$, $x=b \to u=g(b)$. Convert IMMEDIATELY.
Trig Identities
| $\sin^2 x + \cos^2 x = 1$ | $1 + \tan^2 x = \sec^2 x$ | $1 + \cot^2 x = \csc^2 x$ |
| $\sin^2 u = \frac{1-\cos 2u}{2}$ | $\cos^2 u = \frac{1+\cos 2u}{2}$ | $\sin 2u = 2\sin u\cos u$ |
Limits
| $\tan^{-1}(\infty) = \frac{\pi}{2}$ | $\tan^{-1}(0) = 0$ | $\tan^{-1}(-\infty) = -\frac{\pi}{2}$ | $e^{-\infty} = 0$ | $e^{\infty} = \infty$ |
| $\ln(0^+) = -\infty$ | $\frac{1}{\infty} = 0$ | $\frac{1}{0^+} = \infty$ | $\frac{1}{0^-} = -\infty$ | $e^0 = 1$ |
| $\ln(1) = 0$ | $\lim_{x\to 0^+} x\ln x = 0$ | $\lim_{x\to\infty} \frac{\ln x}{x} = 0$ | $\frac{\sin x}{x} \to 1\ (x{\to}0)$ | $\lim_{x\to\infty} x e^{-x} = 0$ |
Integration Techniques
| Technique | Formula | Notes |
|---|
| IBP | $\int u,dv = uv - \int v,du$ | LIATE: Log > InvTrig > Algebra > Trig > Exp |
| PF: $(x-a)$ | $\frac{A}{x-a} \to A\ln\lvert x-a\rvert$ | Single linear factor |
| PF: $(x-a)^n$ | $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}$ | Each power gets its own term |
| PF: $(x^2+bx+c)$ | $\frac{Ax+B}{x^2+bx+c}$ | Irreducible quadratic → numerator is linear |
| Trig sub ($a^2+x^2$) | $x=a\tan\theta$ | $\sqrt{a^2-x^2} \to x=a\sin\theta$ · $\sqrt{x^2-a^2} \to x=a\sec\theta$ |
| L’Hôpital | $\frac{0}{0}$ or $\frac{\infty}{\infty}$ → $\lim \frac{f’}{g’}$ | Diff top & bottom separately (NOT quotient rule) |
| FTC | $\frac{d}{dx}\int_a^{g(x)} f(t),dt = f(g(x)) \cdot g’(x)$ | Chain rule on upper bound |
| Differentiability | $f’(a) = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h}$ | Piecewise: left & right limits must match |
| Area | $\int_a^b [\text{top} - \text{bottom}],dx$ | Sketch first to identify which is top |
FDT: $f’$ changes $+\to-$ → local max · $-\to+$ → local min · same sign → no extremum. SDT inconclusive ($f’’=0$) → fall back to FDT.
Common Errors
| Error | Fix |
|---|
| Evaluate across interior discontinuity | Scan denominator for zeros → split |
| $\sqrt{\text{negative}}$ = NOT REAL | $\lvert x-a\rvert$ left of $a$ → use $(a-x)$, chain rule $(-1)$ |
| Optimization: max or min? | $f’’ < 0$ → max (concave down) · $f’’ > 0$ → min (concave up) |
| Forgot shifted axis radius | Radius $= f(x) - k$, not $f(x)$ |