Because integration reverses differentiation, every derivative formula gives a corresponding standard integral. The core standard results (each carries + c +c + c
∫ 0 d x = c \int 0\,dx = c ∫ 0 d x = c ∫ k d x = k x + c \int k\,dx = kx+c ∫ k d x = k x + c k k k Power rule: ∫ x n d x = x n + 1 n + 1 + c , ( n ≠ − 1 ) \int x^{n}\,dx = \dfrac{x^{n+1}}{n+1}+c,\ (n\ne -1) ∫ x n d x = n + 1 x n + 1 + c , ( n = − 1 ) ∫ 1 x d x = log ∣ x ∣ + c \int \dfrac1x\,dx = \log|x|+c ∫ x 1 d x = log ∣ x ∣ + c n = − 1 n=-1 n = − 1 ∫ sin x d x = − cos x + c \int \sin x\,dx = -\cos x+c ∫ sin x d x = − cos x + c ∫ cos x d x = sin x + c \int \cos x\,dx = \sin x+c ∫ cos x d x = sin x + c ∫ sec 2 x d x = tan x + c \int \sec^2 x\,dx = \tan x+c ∫ sec 2 x d x = tan x + c ∫ cosec 2 x d x = − cot x + c \int \operatorname{cosec}^2 x\,dx = -\cot x+c ∫ cosec 2 x d x = − cot x + c ∫ sec x tan x d x = sec x + c \int \sec x\tan x\,dx = \sec x+c ∫ sec x tan x d x = sec x + c ∫ cosec x cot x d x = − cosec x + c \int \operatorname{cosec} x\cot x\,dx = -\operatorname{cosec} x+c ∫ cosec x cot x d x = − cosec x + c ∫ e x d x = e x + c \int e^{x}\,dx = e^{x}+c ∫ e x d x = e x + c ∫ a x d x = a x log a + c \int a^{x}\,dx = \dfrac{a^{x}}{\log a}+c ∫ a x d x = log a a x + c ∫ 1 1 − x 2 d x = sin − 1 x + c \int \dfrac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1}x+c ∫ 1 − x 2 1 d x = sin − 1 x + c ∫ 1 1 + x 2 d x = tan − 1 x + c \int \dfrac{1}{1+x^2}\,dx = \tan^{-1}x+c ∫ 1 + x 2 1 d x = tan − 1 x + c