{
 "source": "https://peabody.kids/flashcards/math-integral-rules",
 "id": "math-integral-rules",
 "title": "Integral Rules and Antiderivatives",
 "category": "math",
 "subject": "Math",
 "levels": [
  "college",
  "9-12"
 ],
 "description": "Antiderivatives of common functions, integration techniques and the fundamental theorem of calculus.",
 "tags": [
  "calculus",
  "integrals"
 ],
 "frontLabel": "Integral",
 "backLabel": "Result",
 "cards": [
  [
   "∫ 0 dx",
   "C"
  ],
  [
   "∫ k dx",
   "kx + C"
  ],
  [
   "∫ xⁿ dx (n ≠ −1)",
   "xⁿ⁺¹/(n + 1) + C"
  ],
  [
   "∫ 1/x dx",
   "ln|x| + C"
  ],
  [
   "∫ eˣ dx",
   "eˣ + C"
  ],
  [
   "∫ aˣ dx",
   "aˣ/ln a + C"
  ],
  [
   "∫ sin x dx",
   "−cos x + C"
  ],
  [
   "∫ cos x dx",
   "sin x + C"
  ],
  [
   "∫ sec² x dx",
   "tan x + C"
  ],
  [
   "∫ csc² x dx",
   "−cot x + C"
  ],
  [
   "∫ sec x tan x dx",
   "sec x + C"
  ],
  [
   "∫ csc x cot x dx",
   "−csc x + C"
  ],
  [
   "∫ tan x dx",
   "−ln|cos x| + C"
  ],
  [
   "∫ cot x dx",
   "ln|sin x| + C"
  ],
  [
   "∫ sec x dx",
   "ln|sec x + tan x| + C"
  ],
  [
   "∫ ln x dx",
   "x ln x − x + C"
  ],
  [
   "∫ 1/(1 + x²) dx",
   "arctan x + C"
  ],
  [
   "∫ 1/√(1 − x²) dx",
   "arcsin x + C"
  ],
  [
   "∫ x² dx",
   "x³/3 + C"
  ],
  [
   "∫ x³ dx",
   "x⁴/4 + C"
  ],
  [
   "∫ √x dx",
   "(2/3) x^(3/2) + C"
  ],
  [
   "∫ 1/x² dx",
   "−1/x + C"
  ],
  [
   "∫ e^(2x) dx",
   "e^(2x)/2 + C"
  ],
  [
   "∫ cos(3x) dx",
   "sin(3x)/3 + C"
  ],
  [
   "∫ 2x dx",
   "x² + C"
  ],
  [
   "∫ (3x² + 2) dx",
   "x³ + 2x + C"
  ],
  [
   "Constant of integration",
   "+ C, since many functions share the same derivative"
  ],
  [
   "Linearity of integrals",
   "∫ (af + bg) dx = a∫f dx + b∫g dx"
  ],
  [
   "Fundamental theorem of calculus, part 1",
   "d/dx ∫ₐˣ f(t) dt = f(x)"
  ],
  [
   "Fundamental theorem of calculus, part 2",
   "∫ₐᵇ f(x) dx = F(b) − F(a), where F' = f"
  ],
  [
   "Definite integral",
   "The signed area between a curve and the x-axis over an interval"
  ],
  [
   "Indefinite integral",
   "The family of all antiderivatives of a function"
  ],
  [
   "Integration by parts",
   "∫ u dv = uv − ∫ v du"
  ],
  [
   "Substitution (u-substitution)",
   "Let u = g(x), then replace g'(x) dx with du"
  ],
  [
   "Partial fractions",
   "Split a rational function into simpler fractions before integrating"
  ],
  [
   "Riemann sum",
   "A sum of rectangle areas that approximates a definite integral"
  ],
  [
   "Trapezoidal rule",
   "Approximating area using trapezoids instead of rectangles"
  ],
  [
   "Average value of a function",
   "(1/(b − a)) ∫ₐᵇ f(x) dx"
  ],
  [
   "Area between curves",
   "∫ (top − bottom) dx"
  ],
  [
   "Disk method",
   "V = π ∫ [R(x)]² dx"
  ],
  [
   "Washer method",
   "V = π ∫ ([R(x)]² − [r(x)]²) dx"
  ],
  [
   "Shell method",
   "V = 2π ∫ (radius)(height) dx"
  ],
  [
   "Arc length",
   "L = ∫ √(1 + [f'(x)]²) dx"
  ],
  [
   "∫₀¹ x dx",
   "1/2"
  ],
  [
   "∫₀¹ x² dx",
   "1/3"
  ],
  [
   "∫₀^π sin x dx",
   "2"
  ],
  [
   "∫₁^e (1/x) dx",
   "1"
  ],
  [
   "∫₀^1 eˣ dx",
   "e − 1"
  ],
  [
   "Improper integral",
   "An integral with an infinite limit or an unbounded function, defined with a limit"
  ],
  [
   "Position from velocity",
   "s(t) = ∫ v(t) dt"
  ],
  [
   "Velocity from acceleration",
   "v(t) = ∫ a(t) dt"
  ],
  [
   "Even function integral over [−a, a]",
   "2 ∫₀ᵃ f(x) dx"
  ],
  [
   "Odd function integral over [−a, a]",
   "0"
  ],
  [
   "Swapping limits of integration",
   "∫ₐᵇ f = −∫ᵦᵃ f"
  ]
 ]
}