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