Basic Integral Formulas

Almost every integral you will ever be asked to compute is built out of a surprisingly small number of formulas. The list on this page is that list: the integrals of powers, exponentials, logarithms, the six trigonometric functions, their inverses, and the three hyperbolic functions. It is short enough to memorize, and it is worth memorizing, because the techniques that come later are all about recognizing when one of these formulas is hiding inside a harder-looking integrand.

What makes the page more than a formula sheet is the attention paid to conditions. Several of these formulas hold only on part of the real line, and one of them changes form entirely near the ends of its domain. Knowing the formula without knowing where it applies is how confident wrong answers are produced.

Definition

An antiderivative of a function $f$ on an interval $I$ is a function $F$ with $F'(x) = f(x)$ on $I$. The indefinite integral is the family of all antiderivatives of $f$:

$$\int f(x)\,dx = F(x) + C,$$

where $C$ is an arbitrary constant. Two properties of this definition drive everything below. Every formula on this page can be verified rather than remembered: differentiate the right-hand side and you must get the left-hand side back. And because differentiation cannot distinguish two functions differing by a constant, the constant $C$ must be carried along — a formula without a $C$ is incomplete.

The Power Rule

The single most used formula in the subject is the power rule for integrals:

$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \qquad n \neq -1.$$

Special cases worth having instantly available are $\int x\,dx = \frac{x^2}{2} + C$, $\int x^2\,dx = \frac{x^3}{3} + C$, and $\int x^3\,dx = \frac{x^4}{4} + C$.

The rule excludes $n = -1$, because the denominator $n+1$ vanishes there. That case has its own formula, $\int \frac{1}{x}\,dx = \ln\lvert x\rvert + C$, and the absolute value is not optional: $\ln x$ is defined only for $x > 0$, so it cannot be an antiderivative on the negative half-line, while $\ln\lvert x\rvert$ is defined for every $x \neq 0$ and satisfies $\frac{d}{dx}\ln\lvert x\rvert = \frac{1}{x}$ on both halves of the line.

The Exponential and Logarithmic Formulas

$$\int e^x\,dx = e^x + C, \qquad \int a^x\,dx = \frac{a^x}{\ln a} + C \quad (a > 0,\ a \neq 1).$$

The second formula is the first with a change of base: since $\frac{d}{dx}a^x = a^x\ln a$, the factor $\frac{1}{\ln a}$ undoes it. The restriction $a > 0$ keeps $\ln a$ defined, and $a \neq 1$ keeps $\ln a \neq 0$. Examples that follow immediately are $\int e^{2x}\,dx = \frac{1}{2}e^{2x} + C$, $\int 2^x\,dx = \frac{2^x}{\ln 2} + C$, and $\int 3^x\,dx = \frac{3^x}{\ln 3} + C$.

The Trigonometric Formulas

The four basic ones, in the order worth memorizing, and then the two reciprocals:

$$\int \sin x\,dx = -\cos x + C, \qquad \int \cos x\,dx = \sin x + C, \qquad \int \tan x\,dx = -\ln\lvert\cos x\rvert + C,$$ $$\int \cot x\,dx = \ln\lvert\sin x\rvert + C, \qquad \int \sec x\,dx = \ln\lvert\sec x + \tan x\rvert + C, \qquad \int \csc x\,dx = -\ln\lvert\csc x + \cot x\rvert + C.$$

Every one of these can be checked by differentiating. For example, $\frac{d}{dx}(-\ln\lvert\cos x\rvert) = \frac{\sin x}{\cos x} = \tan x$, and $\frac{d}{dx}\ln\lvert\sec x + \tan x\rvert = \frac{\sec x\tan x + \sec^2 x}{\sec x + \tan x} = \sec x$.

The Inverse Trigonometric Formulas

$$\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C, \qquad \int \frac{1}{1+x^2}\,dx = \arctan x + C.$$

The first is valid on $\lvert x\rvert < 1$, where the integrand is real; the second is valid for every real $x$.

A third formula concerns the region $\lvert x\rvert > 1$, and it is easy to get wrong. The correct statement is $\int \frac{1}{\lvert x\rvert\sqrt{x^2-1}}\,dx = \arccos\left(\frac{1}{x}\right) + C$ for $\lvert x\rvert > 1$.

The Hyperbolic Formulas

Their integrals mirror the circular functions:

$$\int \sinh x\,dx = \cosh x + C, \qquad \int \cosh x\,dx = \sinh x + C, \qquad \int \tanh x\,dx = \ln\lvert\cosh x\rvert + C.$$

The last one follows from $\frac{d}{dx}\ln\lvert\cosh x\rvert = \frac{\sinh x}{\cosh x} = \tanh x$. Note that $\cosh x \geq 1$ for all real $x$, so the absolute value in that logarithm is harmless.

How to Use the Table

Group by type. Powers with the power rule, exponentials with the exponential formulas, reciprocals with the logarithmic formula, trigonometric and hyperbolic functions with their own rows, and inverse trigonometric functions with theirs. Identifying the family is most of the work. Then verify by differentiating and read the domain: integration is the inverse of differentiation, so any formula can be checked in one line, and a check catches a lost factor of $\tfrac{1}{2}$ or a missing absolute value immediately. The formulas for $\arcsin x$, $\arctan x$, $\ln\lvert x\rvert$, $\tan x$, $\cot x$, $\sec x$ and $\csc x$ each come with a restriction on where the integrand is defined, and that restriction should be stated when the formula is used.

Exercises

Exercise 1

Compute $\int x^4\,dx$.

Exercise 2

Compute $\int \frac{1}{x^3}\,dx$.

Exercise 3

Compute $\int e^{5x}\,dx$.

Exercise 4

Compute $\int \cos x\,dx$.

Exercise 5

Compute $\int \tan x\,dx$.

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Summary

The catalogue above is small, and that is the point: powers, exponentials, logarithms, the six trigonometric functions, their inverses, and the three hyperbolic functions cover the overwhelming majority of integrals you will meet. The power rule handles every polynomial and every negative power except $-1$, which has its own logarithmic formula; the chain rule contributes the constants that appear in front of the exponential and trigonometric antiderivatives; and the inverse trigonometric formulas carry the domain restrictions that are most often forgotten.

Two habits make the table reliable. Verify every formula by differentiating it, since integration is the inverse of differentiation and the check takes one line. And read the conditions attached to each entry, because the absolute value in $\ln\lvert x\rvert$, the sign in $-\cos x$, and the domain of $\arcsin x$ are not decoration but the substance of the formula. The remaining techniques in this course — substitution, integration by parts, and the rest — are all ways of reshaping an integrand until one of these formulas applies.

Symbols Used in This Article