∫ 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

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