Integral Table
Common indefinite integrals and integration rules — filter and copy.
Click any formula to copy it.
An integral table lists antiderivatives of common functions. For example ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C when n ≠ −1, ∫ eˣ dx = eˣ + C, and ∫ sin x dx = −cos x + C. Every indefinite integral carries a “+ C”, the arbitrary constant of integration.
How to read this table
Integration is the reverse of differentiation: an antiderivative of a function f is any function whose derivative is f. Because the derivative of a constant is zero, every function has infinitely many antiderivatives differing by a constant, so each indefinite integral ends in “+ C”. To check any line, differentiate the right-hand side — you should recover the integrand. For instance, the derivative of −cos x is sin x, which confirms ∫ sin x dx = −cos x + C.
Families of integrals
The basic forms cover powers, reciprocals, exponentials, and logarithms. The trigonometric forms invert the derivatives of the six trig functions. The inverse-trig forms recognise rational and radical expressions whose antiderivatives are arctan or arcsin. The rules — constant multiple, sum, and integration by parts — let you break a complicated integral into these standard pieces.
Basic integrals
Powers, reciprocals, exponentials, and logarithms. Here k and a are constants.
| Integral | Notes |
|---|---|
| ∫ k dx = kx + C | Constant times x |
| ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C | Power rule, valid for n ≠ −1 |
| ∫ (1/x) dx = ln|x| + C | The n = −1 case |
| ∫ eˣ dx = eˣ + C | eˣ is its own antiderivative |
| ∫ aˣ dx = aˣ/ln a + C | For a > 0, a ≠ 1 |
| ∫ ln x dx = x ln x − x + C | Found via integration by parts |
Trigonometric integrals
The antiderivatives of the six trig functions and their standard combinations.
| Integral | Notes |
|---|---|
| ∫ sin x dx = −cos x + C | d/dx(−cos x) = sin x |
| ∫ cos x dx = sin x + C | d/dx(sin x) = cos x |
| ∫ sec²x dx = tan x + C | d/dx(tan x) = sec²x |
| ∫ csc²x dx = −cot x + C | d/dx(−cot x) = csc²x |
| ∫ sec x tan x dx = sec x + C | d/dx(sec x) = sec x tan x |
| ∫ csc x cot x dx = −csc x + C | d/dx(−csc x) = csc x cot x |
| ∫ tan x dx = −ln|cos x| + C | Equivalently ln|sec x| + C |
| ∫ cot x dx = ln|sin x| + C | d/dx(ln|sin x|) = cot x |
| ∫ sec x dx = ln|sec x + tan x| + C | Standard secant integral |
| ∫ csc x dx = −ln|csc x + cot x| + C | Standard cosecant integral |
Inverse-trig forms and rules
Antiderivatives that produce inverse trig functions, plus the linearity and parts rules.
| Integral | Notes |
|---|---|
| ∫ 1/(1 + x²) dx = arctan x + C | d/dx(arctan x) = 1/(1 + x²) |
| ∫ 1/√(1 − x²) dx = arcsin x + C | Valid for −1 < x < 1 |
| ∫ c·f(x) dx = c ∫ f(x) dx | Constant multiple rule |
| ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx | Sum and difference rule |
| ∫ u dv = uv − ∫ v du | Integration by parts |
Why “+ C” is not optional
Leaving off the constant of integration is the most common mistake with indefinite integrals. Because any two antiderivatives of the same function differ only by a constant, the answer is a whole family of functions, not one — writing “+ C” records that. The constant only disappears in a definite integral, where the two constants cancel when you subtract the bounds. Whenever you are unsure of a result, differentiate it: integration and differentiation are inverse operations, so a correct antiderivative must give back the original integrand.