Basic rules
Power rule∫ xⁿ dx = xⁿ⁺¹ / (n+1) + Cn ≠ −1; x > 0 for general real n
1 / x∫ (1/x) dx = ln|x| + Cx ≠ 0
Constant multiple∫ c·f(x) dx = c ∫ f(x) dxc = constant
Sum / difference∫ (f ± g) dx = ∫f dx ± ∫g dx
Substitution∫ f(g(x))·g′(x) dx = ∫ f(u) duu = g(x)
By parts∫ u dv = uv − ∫ v du
Trigonometric
x is in radians; each rule holds on any interval where the integrand is defined.
sin x∫ sin x dx = −cos x + C
cos x∫ cos x dx = sin x + C
sec²x∫ sec²x dx = tan x + C
cosec²x∫ cosec²x dx = −cot x + C
sec x tan x∫ sec x · tan x dx = sec x + C
cosec x cot x∫ cosec x · cot x dx = −cosec x + C
Exponential, log & inverse trig
e^x∫ eˣ dx = eˣ + C
a^x∫ aˣ dx = aˣ / ln a + Ca > 0, a ≠ 1
1 / (1 + x²)∫ dx / (1 + x²) = tan⁻¹x + C
1 / √(1 − x²)∫ dx / √(1 − x²) = sin⁻¹x + C|x| < 1
Good to know
Constant of integrationAlways add + C for indefinite integrals
Definite integral∫ab f(x) dx = F(b) − F(a)F = any antiderivative of f
