Master Integration: Techniques, Formulas and Applications in 136 Flashcards
Learn integration with 136 real flashcards: the standard table, u-substitution, parts, trig substitution, volumes, improper integrals and a four-pass study plan.
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Differentiation is mechanical. Integration is pattern recognition: there is no single algorithm, so the skill is looking at an integrand and knowing within seconds which of six techniques it wants. Integration: Techniques & Applications (deck code AV2951, 136 cards) on Flip It drills exactly that reflex, from the power rule to the Risch algorithm.
Study this deck: Integration: Techniques & Applications
1 / 10Integral
Tap the card for the answer · swipe right if you knew it, left if not
Ten cards from the deck
| Front | Back |
|---|---|
| The integral symbol ∫ | Integration, introduced by Leibniz in 1675, adapted from the long s (ſ) standing for summa (sum) |
| State the fundamental theorem of calculus (part 1) | If F is an antiderivative of continuous f on [a,b], then ∫[a to b] f(x)dx = F(b) − F(a) |
| Power rule for integration | ∫x^n dx = x^(n+1)/(n+1) + C, for n ≠ −1 |
| LIATE rule | Choosing u in integration by parts: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential |
| Find ∫x·e^x dx using integration by parts | xe^x − e^x + C = e^x(x − 1) + C. Let u = x, dv = e^x dx |
| For √(a² − x²), what substitution do you use? | x = a·sin(θ), which gives √(a² − x²) = a·cos(θ) |
| Find ∫ln(x) dx | x·ln(x) − x + C. Integration by parts with u = ln(x), dv = dx |
| If f is odd, what is ∫[−a to a] f(x)dx? | 0 — positive and negative areas cancel by symmetry |
| Gaussian integral | ∫[−∞,∞] e^(−x²)dx = √π |
| Volume of a solid of revolution (shell method) | V = 2π∫[a,b] x·f(x) dx, rotating around the y-axis |
What the 136 cards cover
- Vocabulary: integrand, dx, limits of integration, definite vs indefinite integral, antiderivative, constant of integration C, Riemann sum, Lebesgue integral, method of exhaustion (Eudoxus, Archimedes), plus both parts of the fundamental theorem.
- The standard table: ∫x^n, ∫1/x = ln|x|, ∫e^x, ∫a^x = a^x/ln(a), sin, cos, sec²x, csc²x, sec x tan x, csc x cot x, 1/(1 + x²) = arctan x, 1/√(1 − x²) = arcsin x, plus the logarithmic results for tan, cot, sec and csc.
- Techniques: u-substitution, integration by parts, partial fraction decomposition, the three trigonometric substitutions, completing the square.
- Properties: linearity, constant multiple, sum and difference, reversed limits, ∫[a,a] f = 0, interval additivity, the comparison property, even/odd symmetry, average value, the mean value theorem for integrals.
- Improper integrals: infinite limits, convergence vs divergence, ∫[1,∞] 1/x² dx = 1.
- Applications: signed area, area between curves, disk, washer and shell volumes, arc length, surface area of revolution, displacement vs total distance travelled.
- Numerical methods: trapezoidal rule, Simpson's rule.
- Beyond Calc II: double integrals, Fubini's theorem, line and surface integrals, nonelementary integrals, the error function erf, Liouville's theorem, the Risch algorithm, beta and gamma functions, Cauchy principal value, the Dirichlet integral, ∫[0,π/2] sinⁿx via double factorials.
Work it in four passes
Pass 1 — the table. Cards 21–30 and 97–104 are pure recall and repeat in two notations. Get each under two seconds; every technique card downstream assumes them.
Pass 2 — recognition, not computation. Cover the backs of the technique cards and only answer which method? A product whose second factor hides the derivative of an inner function → u-substitution. Polynomial × transcendental → parts. A radical of the form √(a² ± x²) → trig substitution. A rational function → partial fractions.
Pass 3 — the worked problems (∫2x·cos(x²)dx, ∫x·sin(x)dx, ∫e^x·sin(x)dx, ∫x/(x² + 1)dx) on paper before flipping, then the geometry and special-function cards last.
Two tips specific to integration
Check every answer by differentiating it. Integration is the only topic where the deck hands you a built-in verifier: if d/dx of the back does not return the front, you are wrong. Use it instead of re-reading.
Memorise the n = −1 exception as its own card. The power rule breaks at exactly one exponent, and ∫(1/x)dx = ln|x| + C is where most lost marks live. Card 77 exists for this reason.
Study this deck alongside Derivatives: Rules, Techniques & Applications — the fundamental theorem only clicks once both directions are automatic.
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