Table of Basic Integrals

Basic indefinite integrals of the elementary functions.

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1

\int \frac{d x}{x} = ln ∣ x ∣ + C

\int a^{x} d x = \frac{a ^{x}}{ln a} + C, a > 0, a \neq = 1

\int e^{x} d x = e^{x} + C

\int cos x d x = sin x + C

\int sin x d x = - cos x + C

\int \frac{d x}{cos ^{2} x} = tan x + C

\int \frac{d x}{sin ^{2} x} = - cot x + C

\int \frac{d x}{a ^{2} - x ^{2}} = arcsin \frac{x}{a} + C, a > 0

\int \frac{d x}{a ^{2} + x ^{2}} = \frac{1}{a} \cdot arctan \frac{x}{a} + C, a \neq = 0

\int \frac{d x}{x ^{2} - a ^{2}} = \frac{1}{2 a} \cdot ln \frac{x - a}{x + a} + C, a \neq = 0

\int \frac{d x}{x ^{2} \pm a ^{2}} = ln x + x^{2} \pm a^{2} + C, a \neq = 0 (*)

\int \frac{d x}{sin x} = ln tan \frac{x}{2} + C (**)

\int \frac{d x}{cos x} = ln tan (\frac{x}{2} + \frac{π}{4}) + C

\int cosh x d x = sinh x + C

\int sinh x d x = cosh x + C

\int \frac{d x}{cosh ^{2} x} = tanh x + C

\int \frac{d x}{sinh ^{2} x} = - coth x + C

\int a^{x} d x = \frac{1}{ln a} a^{x} + C

\int ln x d x = x ln x - x + C

\int tan x d x = ln ∣ sec x ∣ + C

\int sec^{2} x d x = tan x + C

\int sec x tan x d x = sec x + C

\int sec x d x = ln ∣ sec x + tan x ∣ + C

\int \frac{a}{a ^{2} + x ^{2}} d x = arctan \frac{x}{a} + C

\int \frac{1}{a ^{2} - x ^{2}} d x = arcsin \frac{x}{a} + C

\int x^{n} d x = \frac{1}{n + 1} x^{n + 1} + C, n \neq = - 1

\int \frac{1}{x} d x = ln ∣ x ∣ + C

\int u d v = uv - \int v d u

\int \frac{1}{a x + b} d x = \frac{1}{a} ln ∣ a x + b ∣ + C

\int x^{n} d x = \frac{1}{n + 1} x^{n + 1} + C, n \neq = - 1

\int \frac{a}{x x ^{2} - a ^{2}} d x = arcsec \frac{x}{a} + C

\int \frac{a}{a ^{2} - x ^{2}} d x = \frac{1}{2} ln \frac{x + a}{x - a} + C

\int \frac{1}{x ^{2} + a ^{2}} d x = sinh^{- 1} \frac{x}{a} = ln (x + x^{2} + a^{2}) + C

\int \frac{1}{x ^{2} - a ^{2}} d x = cosh^{- 1} \frac{x}{a} = ln (x + x^{2} - a^{2}) + C