Table of Integrals: Standard Formulas & Examples

What Is A Table Of Integrals?

A Table of Integrals is a reference list that pairs each standard function with its antiderivative, grouped by family so you can find a form at a glance. It is the integration counterpart of a table of derivatives: where the derivative table tells you how to go from a function to its rate of change, the integral table tells you how to go back the other way, from a rate to the function that produced it.

Two facts govern every row:

• Each indefinite integral ends in $+C$. A function has infinitely many antiderivatives, all differing by a constant, so the general answer to an indefinite integral always adds an arbitrary constant of integration. See constant of integration for why that constant is never optional.

• Each row is a derivative fact read backwards. Because integration reverses differentiation, you can confirm any entry by differentiating the right-hand side. If it returns the integrand, the row is correct.

Throughout this table $a$ is a positive constant, $C$ is the constant of integration, and $n$ is a real number. For the full catalogue of specialised forms beyond a first course, the extended list of integrals collects the rarer cases.

How Do You Read And Use A Table Of Integrals?

Reading a row is a two-step move: match the integrand to a left-hand column, then copy the right-hand column and attach $+C$. The skill is pattern-matching, not memorising every result.

Take a plain power. To evaluate $\int x^5,dx$, match it to the power row $\int x^n,dx = \frac{x^{n+1}}{n+1} + C$ with $n = 5$:

$$\int x^5,dx = \frac{x^{5+1}}{5+1} + C = \frac{x^6}{6} + C$$

Check it by differentiating back: $\frac{d}{dx}\left(\frac{x^6}{6}\right) = \frac{6x^5}{6} = x^5$, the original integrand. The row holds.

The same row handles negative and fractional powers. For $\int \frac{1}{x^3},dx$, rewrite the integrand as $x^{-3}$ and apply $n = -3$:

$$\int x^{-3},dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$$

Differentiating $-\frac{1}{2x^2} = -\frac{1}{2}x^{-2}$ gives $-\frac{1}{2}(-2)x^{-3} = x^{-3}$, so this is correct too. The one power the row cannot touch is $n = -1$, which has its own row below.

What Are The Basic Power And Exponential Integrals?

These are the rows you reach for most often. Every entry is an indefinite integral, so every entry carries $+C$.

Table: Power, reciprocal, and exponential integrals.

The second row is the exception the power rule cannot cover, because $n = -1$ would divide by zero. Its antiderivative is the natural logarithm of the absolute value, $\ln\lvert x\rvert$, and the absolute-value bars matter: $\ln x$ alone is undefined for negative $x$. The basic integration formulas page works through each of these rows one at a time.

What Are The Trigonometric Integrals?

Six rows come straight from reversing the derivatives of the trig functions, and four more (the integrals of $\tan$, $\cot$, $\sec$, $\csc$) need a small logarithm result.

Table: Trigonometric integrals.

The sign on the first row is the one students most often drop: the integral of $\sin x$ is $-\cos x + C$, not $+\cos x$. Verify it the same way as any row, $\frac{d}{dx}(-\cos x) = \sin x$, and the minus sign explains itself. For the derivations of the four log-form rows and products of trig powers, see integration of trigonometric functions.

What Are The Inverse-Trigonometric And Hyperbolic Integrals?

These rows run the other way: the integrand is an algebraic expression, and the antiderivative is an inverse-trig or hyperbolic function. They are the reason a fraction with a square root can suddenly produce an $\arcsin$ or an $\arctan$.

Table: Integrals that produce inverse-trig functions, and the hyperbolic pair.

Note the hyperbolic pair has no minus sign anywhere, unlike its circular cousins: $\int \sinh x,dx = \cosh x + C$ and $\int \cosh x,dx = \sinh x + C$, because $\frac{d}{dx}\cosh x = \sinh x$ with no sign flip. A fuller treatment of $\sinh$, $\cosh$, and their inverses lives on the hyperbolic functions page.

A quick lookup with the arctan row. To evaluate $\int \frac{dx}{9 + x^2}$, match it to $\int \frac{dx}{a^2 + x^2}$ with $a^2 = 9$, so $a = 3$:

$$\int \frac{dx}{9 + x^2} = \frac{1}{3}\arctan\frac{x}{3} + C$$

Check: $\frac{d}{dx}\left(\frac{1}{3}\arctan\frac{x}{3}\right) = \frac{1}{3}\cdot\frac{1/3}{1 + x^2/9} = \frac{1}{9 + x^2}$, the integrand. Likewise $\int \frac{dx}{\sqrt{4 - x^2}} = \arcsin\frac{x}{2} + C$ by the first row with $a = 2$.

What Are The Standard Quadratic Forms?

A few forms involving $a^2$ and $x^2$ come up so often they earn their own rows. They are the workhorses behind trigonometric substitution and integration by partial fractions.

Table: Standard forms with a constant $a > 0$.

The first row is what a partial-fraction split of $\frac{1}{x^2 - a^2}$ produces after you integrate each piece; the second is the area result you would otherwise reach through a trig substitution. Both can be verified by differentiating the right-hand side, and the second is a good exercise in the product rule. For the complete extended catalogue of these forms, the integration formulas reference lists many more.

Why Does A Table Of Integrals Work?

A table of integrals is not a list of coincidences. It works because integration is the exact reverse of differentiation, so every row is a differentiation fact written from right to left.

• The table is a derivative table, flipped. You already know $\frac{d}{dx}\sin x = \cos x$. Reverse the arrow and you have $\int \cos x,dx = \sin x + C$. Every entry in the integral table is some line of the derivative table read backwards, which is why the two references mirror each other row for row.

• Geometry sits behind each row. An indefinite integral is the family of functions whose slope at every point matches the integrand, and a definite version measures the signed area under the curve between two limits. The antiderivative in the right-hand column is the accumulation of that area, and its derivative, the slope, returns you to the integrand.

• The $+C$ records what differentiation forgot. Differentiating erases any constant, since the derivative of a constant is zero. Reversing the process cannot recover which constant was there, so the table restores every one of them at once by adding $+C$.

Because each row is reversible, you are never asked to trust the table blindly. Differentiate the right-hand side and the integrand comes back, every time.

Who Created The First Tables Of Integrals?

Tables of integrals are almost as old as calculus, because the first thing mathematicians did after learning to integrate was to write the results down so no one had to redo them.

Two centuries later the tradition reached its peak in print:

• Izrail Gradshteyn (1899–1958, Russia) and Iosif Ryzhik compiled the Table of Integrals, Series, and Products, a reference of thousands of results that engineers and physicists still keep on the shelf today.

• The modern digital heir is the NIST Digital Library of Mathematical Functions, which puts the same catalogue online with each formula linked to its derivation, closing the loop Euler began.

Where Is A Table Of Integrals Used In The Real World?

The lookup habit reaches well past the calculus classroom, because integrals show up wherever a total is built from a rate.

• Physics and engineering: finding displacement from a velocity function, or the work done by a variable force, means integrating, and practitioners match the integrand to a standard form rather than deriving it afresh.

• Signal processing: transforms that filter and compress audio or images rest on integrals of trig and exponential functions, exactly the rows in the trigonometric and exponential families above.

• Probability and statistics: the area under a probability density is an integral, and standard densities are integrated against the same inverse-trig and exponential forms tabulated here.

• Economics: recovering total cost from marginal cost, or consumer surplus from a demand curve, is a definite integral evaluated with an antiderivative pulled from the table.

• Computer algebra systems: tools like symbolic integrators encode a vast internal table of forms and match your input against it, automating the same lookup a student does by hand.

One reference sheet quietly underwrites motion, sound, risk, and markets, because the act of totalling a rate is the same mathematics everywhere it appears.

What Are The Most Common Mistakes With A Table Of Integrals?

These four errors account for most lost marks when students first work from an integral table, and each matches a question real learners ask on r/calculus, r/learnmath, and course common-error handouts such as Paul's Online Math Notes and MIT OpenCourseWare.

Forgetting the $+C$ on an indefinite integral.

Where it slips in:

A student copies the antiderivative straight off the table and stops, writing $\int \cos x,dx = \sin x$ with no constant.

Don't do this:

Do not treat the right-hand column as the whole answer. An indefinite integral has infinitely many antiderivatives, and dropping $+C$ names only one of them.

The correct way:

Always attach the constant of integration: $\int \cos x,dx = \sin x + C$. The $+C$ is part of the answer, not decoration.

Applying the power rule at $n = -1$.

Where it slips in:

A student integrates $\frac{1}{x} = x^{-1}$ with the power row and writes $\frac{x^{0}}{0} + C$, dividing by zero.

Don't do this:

Do not use $\int x^n,dx = \frac{x^{n+1}}{n+1} + C$ when $n = -1$. The formula is undefined there.

The correct way:

Switch to the exception row: $\int \frac{1}{x},dx = \ln\lvert x\rvert + C$. It is the one power the general rule cannot handle.

Losing the sign on the integral of sine.

Where it slips in:

A student remembers that sine and cosine trade places under integration but writes $\int \sin x,dx = \cos x + C$, keeping the sign positive.

Don't do this:

Do not assume the swap is sign-free. Reversing $\frac{d}{dx}\cos x = -\sin x$ introduces a minus.

The correct way:

Write $\int \sin x,dx = -\cos x + C$, and confirm it by differentiating: $\frac{d}{dx}(-\cos x) = \sin x$.

Dropping the absolute value in a logarithm.

Where it slips in:

A student writes $\int \frac{1}{x},dx = \ln x + C$, which is undefined for negative $x$.

Don't do this:

Do not omit the bars. $\ln x$ only exists for $x > 0$, so it cannot be the antiderivative on the whole domain.

The correct way:

Keep the absolute value: $\int \frac{1}{x},dx = \ln\lvert x\rvert + C$. The same care applies to the log-form trig rows, such as $\int \tan x,dx = \ln\lvert \sec x\rvert + C$.

Practice Problems On The Table Of Integrals

Match each integrand to a row above, then verify your antiderivative by differentiating it back. Answers are verified.

1. $\displaystyle\int x^7,dx$.
   (Answer: $\dfrac{x^8}{8} + C$.)

2. $\displaystyle\int \frac{5}{x},dx$.
   (Answer: $5\ln\lvert x\rvert + C$.)

3. $\displaystyle\int e^x,dx$.
   (Answer: $e^x + C$.)

4. $\displaystyle\int \sec^2 x,dx$.
   (Answer: $\tan x + C$.)

5. $\displaystyle\int \frac{dx}{16 + x^2}$.
   (Answer: $\dfrac{1}{4}\arctan\dfrac{x}{4} + C$, using $a = 4$.)

6. $\displaystyle\int \frac{dx}{\sqrt{25 - x^2}}$.
   (Answer: $\arcsin\dfrac{x}{5} + C$, using $a = 5$.)

Where Should You Go Next After The Table Of Integrals?

The table is a reference, and it points naturally to the techniques that let you use it on integrals that do not match a row on sight.

1. Methods of integration. The map of every reshaping technique, so you know which tool turns a messy integrand into a table form.

2. Integration by substitution. The first technique to learn, for integrands hiding an inner function and its derivative.

3. Integration by parts. The method for products, and the way many log-form and inverse-trig rows are derived.

4. Indefinite integrals. A deeper look at the $+C$ family and what an antiderivative really represents.

If your child is building integration from the ground up, a live Bhanzu trainer teaches the table alongside the reasoning behind each row in the Bhanzu math program.