What Is A List Of Integrals?
A List of Integrals is a reference table pairing each standard function with its antiderivative, the function whose derivative gives you back what you started with. Integration reverses differentiation, so every row in the table is a differentiation fact read backwards. If $\dfrac{d}{dx}F(x) = f(x)$, then
$$\int f(x),dx = F(x) + C.$$
Two things in that line matter. The integrand $f(x)$ is what you are integrating. The $+C$ is the constant of integration: because the derivative of any constant is zero, a function has not one antiderivative but a whole family of them, each shifted vertically. Geometrically, $\int f(x),dx$ measures the signed area building up under the graph of $f$, and $+C$ marks where you chose to start counting that area.
Every entry below is an indefinite integral, so each carries $+C$. Throughout, $a$, $b$, $k$, and $n$ are constants, and $\dfrac{d}{dx}$ is used as the differentiation operator when a formula is checked in reverse.
How Do You Read And Use An Integral Table?
Using the table is a three-step lookup. First, look at your integrand and name its family, is it a power of $x$, an exponential, a trig function, or a fraction with $a^2+x^2$ underneath. Second, find the row whose left side matches that shape, adjusting the constants $a$, $n$, or $k$ to fit your problem. Third, copy the antiderivative and add $+C$.
When the integrand does not match a row exactly, one small rewrite usually lands it on one. The tools that do this rewriting live in their own guides:
Pull out constants and split sums: $\int \big(3x^2 + 4\big),dx = 3\int x^2,dx + 4\int 1,dx$. The methods of integration overview collects these first moves.
Substitute to simplify a composite integrand with integration by substitution.
Split a product with integration by parts when no single row fits.
A quick sanity check settles any row: differentiate the antiderivative. If $\dfrac{d}{dx}F(x)$ returns your integrand, the row is right. That check is how every formula on this page was verified, and it is the fastest way to catch a copying slip in an exam.
What Are The Basic Power And Polynomial Integrals?
These four cover constants, powers of $x$, the reciprocal, and simple linear insides. They handle every polynomial you will meet.
Table: Basic power and polynomial integrals, each an indefinite integral with $+C$.
Integral | Result |
|---|---|
$\displaystyle\int k,dx$ | $kx + C$ |
$\displaystyle\int x^{n},dx\ \ (n \neq -1)$ | $\dfrac{x^{n+1}}{n+1} + C$ |
$\displaystyle\int \dfrac{1}{x},dx$ | $\ln\lvert x\rvert + C$ |
$\displaystyle\int (ax+b)^{n},dx\ \ (n \neq -1)$ | $\dfrac{(ax+b)^{n+1}}{a,(n+1)} + C$ |
Verify the power rule by differentiating its result: $\dfrac{d}{dx}\left(\dfrac{x^{n+1}}{n+1}\right) = \dfrac{(n+1)x^{n}}{n+1} = x^{n}$. The condition $n \neq -1$ is not decoration, at $n = -1$ the formula would divide by zero, and that single missing case is exactly why $\int \dfrac{1}{x},dx$ needs its own row, $\ln\lvert x\rvert + C$. The absolute value keeps the answer valid for negative $x$, since the logarithm of a negative number is undefined.
What Are The Exponential And Logarithmic Integrals?
The exponential $e^x$ is its own antiderivative, which is what makes it the natural base for calculus.
Table: Exponential and logarithmic integrals.
Integral | Result |
|---|---|
$\displaystyle\int e^{x},dx$ | $e^{x} + C$ |
$\displaystyle\int e^{ax},dx$ | $\dfrac{e^{ax}}{a} + C$ |
$\displaystyle\int a^{x},dx\ \ (a>0,\ a \neq 1)$ | $\dfrac{a^{x}}{\ln a} + C$ |
$\displaystyle\int \ln x,dx$ | $x\ln x - x + C$ |
Check the last row, the one students least expect: $\dfrac{d}{dx}\big(x\ln x - x\big) = \ln x + x\cdot\dfrac{1}{x} - 1 = \ln x$. The product rule supplies the $\ln x + 1$, and the $-1$ cancels the extra term, leaving exactly $\ln x$. For the base-$a$ row, $\dfrac{d}{dx}\left(\dfrac{a^{x}}{\ln a}\right) = \dfrac{a^{x}\ln a}{\ln a} = a^{x}$. If you need the rules behind these logarithms first, the logarithm rules page lays them out.
What Are The Trigonometric Integrals?
The six direct trig integrals come straight from reversing the trig derivatives. The four that produce a logarithm ($\tan$, $\cot$, $\sec$, $\csc$) are the ones worth memorising deliberately.
Table: Trigonometric integrals.
Integral | Result |
|---|---|
$\displaystyle\int \sin x,dx$ | $-\cos x + C$ |
$\displaystyle\int \cos x,dx$ | $\sin x + C$ |
$\displaystyle\int \sec^{2} x,dx$ | $\tan x + C$ |
$\displaystyle\int \csc^{2} x,dx$ | $-\cot x + C$ |
$\displaystyle\int \sec x\tan x,dx$ | $\sec x + C$ |
$\displaystyle\int \csc x\cot x,dx$ | $-\csc x + C$ |
$\displaystyle\int \tan x,dx$ | $\ln\lvert\sec x\rvert + C$ |
$\displaystyle\int \cot x,dx$ | $\ln\lvert\sin x\rvert + C$ |
$\displaystyle\int \sec x,dx$ | $\ln\lvert\sec x + \tan x\rvert + C$ |
$\displaystyle\int \csc x,dx$ | $-\ln\lvert\csc x + \cot x\rvert + C$ |
The $\sec x$ row looks like a trick, so check it: $\dfrac{d}{dx}\ln\lvert\sec x + \tan x\rvert = \dfrac{\sec x\tan x + \sec^{2}x}{\sec x + \tan x} = \dfrac{\sec x,(\tan x + \sec x)}{\sec x + \tan x} = \sec x$. The numerator factors to cancel the denominator exactly. For the derivations of the four log-producing rows and more worked cases, see the integration of trigonometric functions guide.
Which Integrals Give Inverse Trigonometric Functions?
These are the patterns to recognise by their denominators. A sum $a^2 + x^2$ points to $\arctan$; a root $\sqrt{a^2 - x^2}$ points to $\arcsin$. Getting fluent at spotting these shapes is most of the skill.
Table: Integrals producing inverse-trigonometric and logarithmic results (here $a > 0$).
Integral | Result |
|---|---|
$\displaystyle\int \dfrac{dx}{a^{2}+x^{2}}$ | $\dfrac{1}{a}\arctan\dfrac{x}{a} + C$ |
$\displaystyle\int \dfrac{dx}{\sqrt{a^{2}-x^{2}}}$ | $\arcsin\dfrac{x}{a} + C$ |
$\displaystyle\int \dfrac{dx}{x\sqrt{x^{2}-a^{2}}}$ | $\dfrac{1}{a}\operatorname{arcsec}\left\lvert\dfrac{x}{a}\right\rvert + C$ |
$\displaystyle\int \dfrac{dx}{a^{2}-x^{2}}$ | $\dfrac{1}{2a}\ln\left\lvert\dfrac{a+x}{a-x}\right\rvert + C$ |
$\displaystyle\int \dfrac{dx}{x^{2}-a^{2}}$ | $\dfrac{1}{2a}\ln\left\lvert\dfrac{x-a}{x+a}\right\rvert + C$ |
$\displaystyle\int \dfrac{dx}{\sqrt{x^{2}+a^{2}}}$ | $\ln\left\lvert x+\sqrt{x^{2}+a^{2}}\right\rvert + C$ |
$\displaystyle\int \dfrac{dx}{\sqrt{x^{2}-a^{2}}}$ | $\ln\left\lvert x+\sqrt{x^{2}-a^{2}}\right\rvert + C$ |
Verify the $\arctan$ row: $\dfrac{d}{dx}\left(\dfrac{1}{a}\arctan\dfrac{x}{a}\right) = \dfrac{1}{a}\cdot\dfrac{1/a}{1+(x/a)^{2}} = \dfrac{1/a^{2}}{(a^{2}+x^{2})/a^{2}} = \dfrac{1}{a^{2}+x^{2}}$. And the $\arcsin$ row: $\dfrac{d}{dx}\arcsin\dfrac{x}{a} = \dfrac{1/a}{\sqrt{1-(x/a)^{2}}} = \dfrac{1/a}{\sqrt{a^{2}-x^{2}}/a} = \dfrac{1}{\sqrt{a^{2}-x^{2}}}$. The two logarithmic rows come from splitting the fraction into partial fractions, and reduce to the same $\ln$ pattern. The $\arctan$ and $\arcsin$ outputs connect back to the arcsin function and its family.
What Are The Hyperbolic Integrals?
The hyperbolic functions integrate almost exactly like the ordinary trig functions, but with none of the sign surprises: $\int \sinh x,dx = \cosh x + C$ has no minus sign, unlike its circular cousin. Reference pages often omit this group, which is why we include it.
Table: Hyperbolic integrals.
Integral | Result |
|---|---|
$\displaystyle\int \sinh x,dx$ | $\cosh x + C$ |
$\displaystyle\int \cosh x,dx$ | $\sinh x + C$ |
$\displaystyle\int \operatorname{sech}^{2} x,dx$ | $\tanh x + C$ |
$\displaystyle\int \operatorname{csch}^{2} x,dx$ | $-\coth x + C$ |
$\displaystyle\int \operatorname{sech} x\tanh x,dx$ | $-\operatorname{sech} x + C$ |
$\displaystyle\int \operatorname{csch} x\coth x,dx$ | $-\operatorname{csch} x + C$ |
Check the first row against the definition $\cosh x = \dfrac{e^{x}+e^{-x}}{2}$: $\dfrac{d}{dx}\cosh x = \dfrac{e^{x}-e^{-x}}{2} = \sinh x$, so $\int \sinh x,dx = \cosh x + C$ as listed. Because $\cosh$ and $\sinh$ are built from $e^{x}$, every hyperbolic integral is really an exponential integral in disguise.
What About Integrals With No Formula? (Non-Elementary Cases)
Not every function has an antiderivative you can write with the standard toolkit, and an honest List of Integrals says so. Two famous integrals look simple but have no elementary closed form:
$\displaystyle\int e^{-x^{2}},dx$ cannot be written with powers, roots, logs, exponentials, or trig functions. It is packaged as the error function, $\operatorname{erf}(x)$, which is defined by this very integral.
$\displaystyle\int \dfrac{\sin x}{x},dx$ has no elementary form either; it is named the sine integral, $\operatorname{Si}(x)$.
These are not gaps in your knowledge, they are theorems: it is proven that no finite combination of standard functions differentiates to $e^{-x^{2}}$. When you meet one, you switch to a definite integral and evaluate it numerically rather than hunting for a formula that does not exist.
How Do You Apply The List Of Integrals? Worked Examples
Each example matches the integrand to a table row, adjusts the constants, and checks the answer by differentiating back.
Example 1: A polynomial (power rule, term by term).
Evaluate $\displaystyle\int \big(3x^{2} + 4x - 5\big),dx$.
Integrate each term with the power rule, and integrate the constant as $kx$:
$$\int \big(3x^{2} + 4x - 5\big),dx = \frac{3x^{3}}{3} + \frac{4x^{2}}{2} - 5x + C = x^{3} + 2x^{2} - 5x + C$$
Check: $\dfrac{d}{dx}\big(x^{3} + 2x^{2} - 5x\big) = 3x^{2} + 4x - 5$, the original integrand.
Final answer: $x^{3} + 2x^{2} - 5x + C$.
Example 2: An inverse-tangent pattern.
Evaluate $\displaystyle\int \dfrac{dx}{9 + x^{2}}$.
The denominator is $a^{2} + x^{2}$ with $a^{2} = 9$, so $a = 3$. The $\arctan$ row gives:
$$\int \frac{dx}{9 + x^{2}} = \frac{1}{3}\arctan\frac{x}{3} + C$$
Check: $\dfrac{d}{dx}\left(\dfrac{1}{3}\arctan\dfrac{x}{3}\right) = \dfrac{1}{3}\cdot\dfrac{1/3}{1+(x/3)^{2}} = \dfrac{1}{9 + x^{2}}$.
Final answer: $\dfrac{1}{3}\arctan\dfrac{x}{3} + C$.
Example 3: An inverse-sine pattern.
Evaluate $\displaystyle\int \dfrac{dx}{\sqrt{25 - x^{2}}}$.
The root is $\sqrt{a^{2} - x^{2}}$ with $a^{2} = 25$, so $a = 5$. The $\arcsin$ row gives:
$$\int \frac{dx}{\sqrt{25 - x^{2}}} = \arcsin\frac{x}{5} + C$$
Check: $\dfrac{d}{dx}\arcsin\dfrac{x}{5} = \dfrac{1/5}{\sqrt{1-(x/5)^{2}}} = \dfrac{1}{\sqrt{25 - x^{2}}}$.
Final answer: $\arcsin\dfrac{x}{5} + C$.
Example 4: Mixing exponential and trig rows.
Evaluate $\displaystyle\int \big(2e^{x} + \sec^{2} x\big),dx$.
Split the sum, then read one row for each piece:
$$\int \big(2e^{x} + \sec^{2} x\big),dx = 2e^{x} + \tan x + C$$
Check: $\dfrac{d}{dx}\big(2e^{x} + \tan x\big) = 2e^{x} + \sec^{2} x$, the integrand.
Final answer: $2e^{x} + \tan x + C$.
Why Does Every Indefinite Integral Need The Constant Of Integration?
The $+C$ is the part beginners drop most often, and it is not a formality. It follows directly from what differentiation throws away.
A whole family, not one function. If $F(x)$ is an antiderivative of $f(x)$, so is $F(x) + 5$, and so is $F(x) - 100$, because differentiating any constant gives zero. The integral names all of them at once by writing $F(x) + C$.
Geometry of the vertical shift. Every member of the family has the same slope $f(x)$ at each point, so the curves are identical in shape, stacked at different heights. The area interpretation is the same picture: $C$ fixes where the running total of area is set to zero.
Why definite integrals do not need it. A definite integral subtracts $F(b) - F(a)$, and the $+C$ cancels itself, which is a direct consequence of the fundamental theorem of calculus. The constant only lives on indefinite integrals.
Reading the table backwards makes the point concrete. Each row is a derivative fact in reverse, and since differentiation erased the original constant, integration has to restore an unknown one. The table gives you $F(x)$; you supply the $+C$.
Who Created The First List Of Integrals?
Tables of integrals are almost as old as calculus itself, assembled by the same mathematicians who were inventing the subject.
By the twentieth century these efforts had grown into reference works of thousands of entries. The best known, the Table of Integrals, Series, and Products first compiled by Izrail Gradshteyn and Iosif Ryzhik (first Russian edition 1943), runs past a thousand pages and sits on the shelf of nearly every physics department. Every printed table also relies on the rigorous definition of the integral given by Bernhard Riemann (1826–1866, Germany), which is what guarantees the areas these formulas compute actually exist.
Where Is The List Of Integrals Used In The Real World?
The table is a working tool wherever a total has to be recovered from a rate.
Physics and engineering: distance from velocity, work from force, and charge from current are all single table lookups once the rate is written as a function of time.
Probability and statistics: the area under a probability density is an integral, and the $\int e^{-x^{2}}$ family sits at the heart of the normal distribution used across data science.
Economics: total cost and total revenue are integrals of the marginal (per-unit) rates a firm can measure directly.
Signal processing and graphics: smoothing a signal or shading a rendered surface accumulates a rate across an interval, which is an integral evaluated numerically at speed.
Engineering design: centres of mass, areas, and volumes of curved parts are standard integrals of the power and root forms in the tables above.
One reference table quietly underwrites motion, risk, money, sound, and structure, which is why every quantitative field keeps a copy close.
What Are The Most Common Mistakes With The List Of Integrals?
These four errors account for most lost marks when working from an integral table, and each matches a correction that appears on university error handouts and r/calculus threads.
Dropping the constant of integration.
Where it slips in:
A student writes $\int 2x,dx = x^{2}$ and stops, treating an indefinite integral like a definite one.
Don't do this:
Do not omit the $+C$. Without it the answer names only one antiderivative out of infinitely many, and is marked wrong.
The correct way:
Always close an indefinite integral with $+C$: $\int 2x,dx = x^{2} + C$. Drop it only for a definite integral, where the constant cancels.
Writing $\ln x$ instead of $\ln\lvert x\rvert$.
Where it slips in:
A student copies $\int \dfrac{1}{x},dx = \ln x + C$ from memory, missing the absolute value.
Don't do this:
Do not drop the bars. $\ln x$ is undefined for negative $x$, so $\ln x + C$ is only half the answer.
The correct way:
Write $\int \dfrac{1}{x},dx = \ln\lvert x\rvert + C$, valid on both sides of zero.
Inventing a product rule for integrals.
Where it slips in:
Faced with $\int x\cos x,dx$, a student writes it as $\big(\int x,dx\big)\big(\int \cos x,dx\big)$.
Don't do this:
Do not integrate a product factor by factor. There is no product rule for integration, and no table row for a general product.
The correct way:
Use integration by parts, which is the reverse of the product rule, to turn the product into pieces the table can handle.
Changing the variable of integration in the answer.
Where it slips in:
Computing $\int \dfrac{1}{2},dt$, a student writes $\dfrac{x}{2} + C$, swapping the integration variable for $x$ out of habit.
Don't do this:
Do not integrate in one variable and answer in another. The result must be in the same variable as $dt$.
The correct way:
Keep the variable: $\int \dfrac{1}{2},dt = \dfrac{t}{2} + C$. Match the answer to whatever follows the $d$.
Practice Problems On The List Of Integrals
Work each one from the tables above, then check against the verified answer.
$\displaystyle\int \big(6x^{2} - 2x + 7\big),dx$.
(Answer: $2x^{3} - x^{2} + 7x + C$.)$\displaystyle\int \dfrac{1}{x^{2}+16},dx$.
(Answer: $\dfrac{1}{4}\arctan\dfrac{x}{4} + C$.)$\displaystyle\int \dfrac{dx}{\sqrt{4 - x^{2}}}$.
(Answer: $\arcsin\dfrac{x}{2} + C$.)$\displaystyle\int \big(e^{2x} + \cos x\big),dx$.
(Answer: $\dfrac{e^{2x}}{2} + \sin x + C$.)$\displaystyle\int \dfrac{5}{x},dx$.
(Answer: $5\ln\lvert x\rvert + C$.)$\displaystyle\int \sinh x,dx$.
(Answer: $\cosh x + C$.)
Where Should You Go Next After The List Of Integrals?
The table is the reference; these topics turn it into skill.
Methods of integration. Learn the rewrites, substitution, parts, and partial fractions, that turn an off-table integrand into an on-table one.
Antiderivatives. Go deeper on what $+C$ means and why one function has a whole family of them.
Definite integrals. Use the table to compute areas and totals with the $F(b) - F(a)$ evaluation.
If your child is learning to integrate for the first time, a live Bhanzu trainer teaches the table alongside the pattern-matching that makes it usable, in the Bhanzu math program.
Was this article helpful?
Your feedback helps us write better content