Formula · Calculus
Integral Formulas: The Complete Table
One short table of base cases, and a second showing what happens to each one when the input is a linear expression instead of a bare x.
The structural rules
These say how to integrate a combination once you can integrate its parts. Every integral formula that follows is applied inside this framework.
| Name | Rule |
|---|---|
| Constant | ∫ k dx = kx + C |
| Constant multiple | ∫ k·u dx = k·∫u dx |
| Sum and difference | ∫ (u ± v) dx = ∫u dx ± ∫v dx |
| Power rule | ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1 |
| The exception | ∫ 1/x dx = ln|x| + C |
The power rule holds for every real exponent except −1, where it would mean dividing by zero. That one case is not a gap — it has its own formula, the logarithm, and together the two cover every power of x. The sum rule is what lets a longer expression be split apart before any of the other rules are applied, one term at a time.
The standard functions: trig, exponential and hyperbolic
Each of these is the antiderivative of a function you likely already know how to differentiate — read the table backward and it is the same reference table used for derivatives. The trig integrals, the exponential row, and the hyperbolic pair below each function are grouped together because they are learned the same way: once, and then looked up rather than re-derived.
| f(x) | ∫ f(x) dx |
|---|---|
| sin x | −cos x + C |
| cos x | sin x + C |
| eˣ | eˣ + C |
| aˣ | aˣ / ln a + C |
| 1/x | ln|x| + C |
| sinh x | cosh x + C |
| cosh x | sinh x + C |
| 1/√x | 2√x + C |
The minus sign on the antiderivative of sin is the one people drop, and the direction is easy to get backward from memory: differentiating −cos x gives sin x back, which is the check worth running if the sign is ever in doubt.
Two forms not on this calculator, for completeness
Two standard integrals are worth knowing even though they need a substitution this calculator does not attempt, since they turn up constantly once integration by parts and trig substitution are introduced.
| f(x) | ∫ f(x) dx |
|---|---|
| 1/(1 + x²) | arctan x + C |
| 1/√(1 − x²) | arcsin x + C |
Both are reverse entries from the inverse-trig derivatives, and both are worth recognising on sight rather than deriving, since neither follows from the power rule or a simple substitution.
The linear-substitution form is the general form
Every entry above is the special case where the input is a bare x. In practice the input is more often a linear expression, ax + b, and the rule is the same antiderivative divided by the coefficient a.
| Base case | With ax + b inside | Example |
|---|---|---|
| ∫ sin x dx = −cos x + C | ∫ sin(ax+b) dx = −cos(ax+b)/a + C | sin(2x) → −cos(2x)/2 |
| ∫ eˣ dx = eˣ + C | ∫ e^(ax+b) dx = e^(ax+b)/a + C | e^(3x) → e^(3x)/3 |
| ∫ 1/x dx = ln|x| + C | ∫ 1/(ax+b) dx = ln|ax+b|/a + C | 1/(2x+1) → ln|2x+1|/2 |
| ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C | ∫ (ax+b)ⁿ dx = (ax+b)ⁿ⁺¹/(a(n+1)) + C | (2x+1)⁵ → (2x+1)⁶/12 |
The division by a exists to cancel the factor the chain rule would otherwise introduce on the way back. Differentiate −cos(2x)/2 and the chain rule brings down a 2, which the division by 2 was placed there to absorb — the check that this table is right, run backward.
Why the divide-by-a step disappears when a = 1
Every row above collapses to its base case when a is 1 and b is 0, since dividing by 1 changes nothing. The base-case table is not a separate set of rules; it is this table with the substitution left out.
Definite integrals and the fundamental theorem
None of the formulas above change when limits are added. Find any antiderivative F using the table, then a definite integral from a to b is F(b) − F(a). The constant of integration is not needed here, because it appears in both terms and cancels exactly.
| Definite integral | Antiderivative | Value |
|---|---|---|
| ∫₀¹ x² dx | x³/3 | ⅓ |
| ∫₀³ 2x dx | x² | 9 |
Questions about integral formulas
What is the formula for integrating x^n?
x^(n+1)/(n+1) + C, for every exponent except −1. When n is −1 the formula would divide by zero, so 1/x integrates to ln|x| + C instead.
What is the antiderivative of sin(ax + b)?
−cos(ax + b)/a + C. The division by a undoes the factor the chain rule would introduce when the result is differentiated back.
Is there a formula for integrating a product of two functions?
Not a direct one — a genuine product of two functions of x needs integration by parts, a separate technique built on the product rule for derivatives run in reverse.
Why does the constant of integration not appear in these tables?
Every antiderivative implicitly carries a + C; it is omitted from the middle of a derivation and restored only in the final answer, since it would otherwise clutter every intermediate line.