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Integral Rules and Antiderivatives
Antiderivatives of common functions, integration techniques and the fundamental theorem of calculus.
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54 cards
- ∫ 0 dxIntegralC
- ∫ k dxIntegralkx + C
- ∫ xⁿ dx (n ≠ −1)Integralxⁿ⁺¹/(n + 1) + C
- ∫ 1/x dxIntegralln|x| + C
- ∫ eˣ dxIntegraleˣ + C
- ∫ aˣ dxIntegralaˣ/ln a + C
- ∫ sin x dxIntegral−cos x + C
- ∫ cos x dxIntegralsin x + C
- ∫ sec² x dxIntegraltan x + C
- ∫ csc² x dxIntegral−cot x + C
- ∫ sec x tan x dxIntegralsec x + C
- ∫ csc x cot x dxIntegral−csc x + C
- ∫ tan x dxIntegral−ln|cos x| + C
- ∫ cot x dxIntegralln|sin x| + C
- ∫ sec x dxIntegralln|sec x + tan x| + C
- ∫ ln x dxIntegralx ln x − x + C
- ∫ 1/(1 + x²) dxIntegralarctan x + C
- ∫ 1/√(1 − x²) dxIntegralarcsin x + C
- ∫ x² dxIntegralx³/3 + C
- ∫ x³ dxIntegralx⁴/4 + C
- ∫ √x dxIntegral(2/3) x^(3/2) + C
- ∫ 1/x² dxIntegral−1/x + C
- ∫ e^(2x) dxIntegrale^(2x)/2 + C
- ∫ cos(3x) dxIntegralsin(3x)/3 + C
- ∫ 2x dxIntegralx² + C
- ∫ (3x² + 2) dxIntegralx³ + 2x + C
- Constant of integrationIntegral+ C, since many functions share the same derivative
- Linearity of integralsIntegral∫ (af + bg) dx = a∫f dx + b∫g dx
- Fundamental theorem of calculus, part 1Integrald/dx ∫ₐˣ f(t) dt = f(x)
- Fundamental theorem of calculus, part 2Integral∫ₐᵇ f(x) dx = F(b) − F(a), where F' = f
- Definite integralIntegralThe signed area between a curve and the x-axis over an interval
- Indefinite integralIntegralThe family of all antiderivatives of a function
- Integration by partsIntegral∫ u dv = uv − ∫ v du
- Substitution (u-substitution)IntegralLet u = g(x), then replace g'(x) dx with du
- Partial fractionsIntegralSplit a rational function into simpler fractions before integrating
- Riemann sumIntegralA sum of rectangle areas that approximates a definite integral
- Trapezoidal ruleIntegralApproximating area using trapezoids instead of rectangles
- Average value of a functionIntegral(1/(b − a)) ∫ₐᵇ f(x) dx
- Area between curvesIntegral∫ (top − bottom) dx
- Disk methodIntegralV = π ∫ [R(x)]² dx
- Washer methodIntegralV = π ∫ ([R(x)]² − [r(x)]²) dx
- Shell methodIntegralV = 2π ∫ (radius)(height) dx
- Arc lengthIntegralL = ∫ √(1 + [f'(x)]²) dx
- ∫₀¹ x dxIntegral1/2
- ∫₀¹ x² dxIntegral1/3
- ∫₀^π sin x dxIntegral2
- ∫₁^e (1/x) dxIntegral1
- ∫₀^1 eˣ dxIntegrale − 1
- Improper integralIntegralAn integral with an infinite limit or an unbounded function, defined with a limit
- Position from velocityIntegrals(t) = ∫ v(t) dt
- Velocity from accelerationIntegralv(t) = ∫ a(t) dt
- Even function integral over [−a, a]Integral2 ∫₀ᵃ f(x) dx
- Odd function integral over [−a, a]Integral0
- Swapping limits of integrationIntegral∫ₐᵇ f = −∫ᵦᵃ f
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