Lists of integrals

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This article is about mostly indefinite integrals in calculus. For a list of definite integrals, see List of definite integrals.

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Part of a series of articles about

Calculus

* Fundamental theorem

* Limits of functions
* Continuity

* Mean value theorem
* Rolle's theorem

Differential

|  Definitions

* Derivative (generalizations)

* Differential
* infinitesimal
* of a function
* total

Concepts

* Differentiation notation
* Second derivative
* Third derivative
* Change of variables
* Implicit differentiation
* Related rates
* Taylor's theorem

Rules and identities

* Sum
* Product
* Chain
* Power
* Quotient
* Inverse
* General Leibniz
* Faà di Bruno's formula

Integral

* Lists of integrals

Definitions

* Antiderivative
* Integral (improper)
* Riemann integral
* Lebesgue integration
* Contour integration

Integration by

* Parts
* Discs
* Cylindrical shells
* Substitution (trigonometric)
* Partial fractions
* Order
* Reduction formulae

Series

* Geometric (arithmetico-geometric)
* Harmonic
* Alternating
* Power
* Binomial
* Taylor

Convergence tests

* Summand limit (term test)
* Ratio
* Root
* Integral
* Direct comparison

Limit comparison
* Alternating series
* Cauchy condensation
* Dirichlet
* Abel

Vector

* Gradient
* Divergence
* Curl
* Laplacian
* Directional derivative
* Identities

Theorems

* Divergence
* Gradient
* Green's
* Kelvin–Stokes
* Stokes

Multivariable

|  Formalisms

* Matrix
* Tensor
* Exterior
* Geometric

Definitions

* Partial derivative
* Multiple integral
* Line integral
* Surface integral
* Volume integral
* Jacobian
* Hessian

Specialized

* Fractional
* Malliavin
* Stochastic
* Variations

Glossary of calculus

* Glossary of calculus

* v
* t
* e

This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries.

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.

## Contents

* 1 Historical development of integrals
* 2 Lists of integrals
* 3 Integrals of simple functions
* 3.1 Integrals with a singularity
* 3.2 Rational functions
* 3.3 Exponential functions
* 3.4 Logarithms
* 3.5 Trigonometric functions
* 3.6 Inverse trigonometric functions
* 3.7 Hyperbolic functions
* 3.8 Inverse hyperbolic functions
* 3.9 Products of functions proportional to their second derivatives
* 3.10 Absolute-value functions
* 3.11 Special functions
* 4 Definite integrals lacking closed-form antiderivatives
* 5 See also
* 6 References
* 7 Further reading
* 8 External links
* 8.1 Tables of integrals
* 8.2 Derivations
* 8.3 Online service
* 8.4 Open source programs
* 8.5 Videos

## Historical development of integrals

A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician Meier Hirsch [de; de] (also spelled Meyer Hirsch) in 1810.[1] These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David Bierens de Haan for his Tables d'intégrales définies, supplemented by Supplément aux tables d'intégrales définies in ca. 1864. A new edition was published in 1867 under the title Nouvelles tables d'intégrales définies.

These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of Gradshteyn and Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by Bierens de Haan are denoted by BI.

Not all closed-form expressions have closed-form antiderivatives; this study forms the subject of differential Galois theory, which was initially developed by Joseph Liouville in the 1830s and 1840s, leading to Liouville's theorem which classifies which expressions have closed-form antiderivatives. A simple example of a function without a closed-form antiderivative is e−x2, whose antiderivative is (up to constants) the error function.

Since 1968 there is the Risch algorithm for determining indefinite integrals that can be expressed in term of elementary functions, typically using a computer algebra system. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the Meijer G-function.

## Lists of integrals

More detail may be found on the following pages for the lists of integrals:

* List of integrals of rational functions
* List of integrals of irrational algebraic functions
* List of integrals of trigonometric functions
* List of integrals of inverse trigonometric functions
* List of integrals of hyperbolic functions
* List of integrals of inverse hyperbolic functions
* List of integrals of exponential functions
* List of integrals of logarithmic functions
* List of integrals of Gaussian functions

Gradshteyn, Ryzhik, Geronimus, Tseytlin, Jeffrey, Zwillinger, and Moll's (GR) Table of Integrals, Series, and Products contains a large collection of results. An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (with volumes 1–3 listing integrals and series of elementary and special functions, volume 4–5 are tables of Laplace transforms). More compact collections can be found in e.g. Brychkov, Marichev, Prudnikov's Tables of Indefinite Integrals, or as chapters in Zwillinger's CRC Standard Mathematical Tables and Formulae or Bronshtein and Semendyayev's Guide Book to Mathematics, Handbook of Mathematics or Users' Guide to Mathematics, and other mathematical handbooks.

Other useful resources include Abramowitz and Stegun and the Bateman Manuscript Project. Both works contain many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. Two volumes of the Bateman Manuscript are specific to integral transforms.

There are several web sites which have tables of integrals and integrals on demand. Wolfram Alpha can show results, and for some simpler expressions, also the intermediate steps of the integration. Wolfram Research also operates another online service, the Mathematica Online Integrator.

## Integrals of simple functions

C is used for an arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus, each function has an infinite number of antiderivatives.

These formulas only state in another form the assertions in the table of derivatives.

### Integrals with a singularity

When there is a singularity in the function being integrated such that the antiderivative becomes undefined at some point (the singularity), then C does not need to be the same on both sides of the singularity. The forms below normally assume the Cauchy principal value around a singularity in the value of C, but this is not necessary in general. For instance, in ∫1xdx=ln⁡|x|+C there is a singularity at 0 and the antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −iπ when using a path above the origin and iπ for a path below the origin. A function on the real line could use a completely different value of C on either side of the origin as in:[2] ∫1xdx=ln⁡|x|+{Aif x>0;Bif x<0.

### Rational functions

See also: List of integrals of rational functions

* ∫adx=ax+C

The following function has a non-integrable singularity at 0 for n ≤ −1:

* ∫xndx=xn+1n+1+C(for n≠−1) (Cavalieri's quadrature formula)
* ∫(ax+b)ndx=(ax+b)n+1a(n+1)+C(for n≠−1)
* ∫1xdx=ln⁡|x|+C
* More generally,[3]∫1xdx={ln⁡|x|+C−x<0ln⁡|x|+C+x>0
* ∫cax+bdx=caln⁡|ax+b|+C

### Exponential functions

See also: List of integrals of exponential functions

* ∫eaxdx=1aeax+C
* ∫f′(x)ef(x)dx=ef(x)+C
* ∫axdx=axln⁡a+C
* ∫ex(f(x)+f′(x))dx=exf(x)+C
* ∫ex(f(x)−(−1)ndnf(x)dxn)dx=ex∑k=1n(−1)k−1dk−1f(x)dxk−1+C

(if n is a positive integer)
* ∫e−x(f(x)−dnf(x)dxn)dx=−e−x∑k=1ndk−1f(x)dxk−1+C

(if n is a positive integer)

### Logarithms

See also: List of integrals of logarithmic functions

* ∫ln⁡xdx=xln⁡x−x+C=x(ln⁡x−1)+C
* ∫logaxdx=xlogax−xln⁡a+C=xln⁡a(ln⁡x−1)+C

### Trigonometric functions

See also: List of integrals of trigonometric functions

* ∫sin⁡xdx=−cos⁡x+C
* ∫cos⁡xdx=sin⁡x+C
* ∫tan⁡xdx=ln⁡|sec⁡x|+C=−ln⁡|cos⁡x|+C
* ∫cot⁡xdx=−ln⁡|csc⁡x|+C=ln⁡|sin⁡x|+C
* ∫sec⁡xdx=ln⁡|sec⁡x+tan⁡x|+C=ln⁡|tan⁡(x2+π4)|+C
* (See Integral of the secant function. This result was a well-known conjecture in the 17th century.)
* ∫csc⁡xdx=−ln⁡|csc⁡x+cot⁡x|+C=ln⁡|csc⁡x−cot⁡x|+C=ln⁡|tan⁡x2|+C
* ∫sec2xdx=tan⁡x+C
* ∫csc2xdx=−cot⁡x+C
* ∫sec⁡xtan⁡xdx=sec⁡x+C
* ∫csc⁡xcot⁡xdx=−csc⁡x+C
* ∫sin2xdx=12(x−sin⁡2x2)+C=12(x−sin⁡xcos⁡x)+C
* ∫cos2xdx=12(x+sin⁡2x2)+C=12(x+sin⁡xcos⁡x)+C
* ∫tan2xdx=tan⁡x−x+C
* ∫cot2xdx=−cot⁡x−x+C
* ∫sec3xdx=12(sec⁡xtan⁡x+ln⁡|sec⁡x+tan⁡x|)+C
* (See integral of secant cubed.)
* ∫csc3xdx=12(−csc⁡xcot⁡x+ln⁡|csc⁡x−cot⁡x|)+C=12(ln⁡|tan⁡x2|−csc⁡xcot⁡x)+C
* ∫sinnxdx=−sinn−1xcos⁡xn+n−1n∫sinn−2xdx
* ∫cosnxdx=cosn−1xsin⁡xn+n−1n∫cosn−2xdx

### Inverse trigonometric functions

See also: List of integrals of inverse trigonometric functions

* ∫arcsin⁡xdx=xarcsin⁡x+1−x2+C, for |x|≤1
* ∫arccos⁡xdx=xarccos⁡x−1−x2+C, for |x|≤1
* ∫arctan⁡xdx=xarctan⁡x−12ln⁡|1+x2|+C, for all real x
* ∫arccot⁡xdx=xarccot⁡x+12ln⁡|1+x2|+C, for all real x
* ∫arcsec⁡xdx=xarcsec⁡x−ln⁡|x(1+1−x−2)|+C, for |x|≥1
* ∫arccsc⁡xdx=xarccsc⁡x+ln⁡|x(1+1−x−2)|+C, for |x|≥1

### Hyperbolic functions

See also: List of integrals of hyperbolic functions

* ∫sinh⁡xdx=cosh⁡x+C
* ∫cosh⁡xdx=sinh⁡x+C
* ∫tanh⁡xdx=ln⁡(cosh⁡x)+C
* ∫coth⁡xdx=ln⁡|sinh⁡x|+C, for x≠0
* ∫sech⁡xdx=arctan⁡(sinh⁡x)+C
* ∫csch⁡xdx=ln⁡|coth⁡x−csch⁡x|+C=ln⁡|tanh⁡x2|+C, for x≠0
* ∫sech2xdx=tanh⁡x+C
* ∫csch2xdx=−coth⁡x+C
* ∫sech⁡xtanh⁡xdx=−sech⁡x+C
* ∫csch⁡xcoth⁡xdx=−csch⁡x+C

### Inverse hyperbolic functions

See also: List of integrals of inverse hyperbolic functions

* ∫arcsinh⁡xdx=xarcsinh⁡x−x2+1+C, for all real x
* ∫arccosh⁡xdx=xarccosh⁡x−x2−1+C, for x≥1
* ∫arctanh⁡xdx=xarctanh⁡x+ln⁡(1−x2)2+C, for |x|<1
* ∫arccoth⁡xdx=xarccoth⁡x+ln⁡(x2−1)2+C, for |x|>1
* ∫arcsech⁡xdx=xarcsech⁡x+arcsin⁡x+C, for 0<x≤1
* ∫arccsch⁡xdx=xarccsch⁡x+|arcsinh⁡x|+C, for x≠0

### Products of functions proportional to their second derivatives

* ∫cos⁡axebxdx=ebxa2+b2(asin⁡ax+bcos⁡ax)+C
* ∫sin⁡axebxdx=ebxa2+b2(bsin⁡ax−acos⁡ax)+C
* ∫cos⁡axcosh⁡bxdx=1a2+b2(asin⁡axcosh⁡bx+bcos⁡axsinh⁡bx)+C
* ∫sin⁡axcosh⁡bxdx=1a2+b2(bsin⁡axsinh⁡bx−acos⁡axcosh⁡bx)+C

### Absolute-value functions

Let f be a continuous function, that has at most one zero. If f has a zero, let g be the unique antiderivative of f that is zero at the root of f; otherwise, let g be any antiderivative of f. Then ∫|f(x)|dx=sgn⁡(f(x))g(x)+C, where sgn(x) is the sign function, which takes the values −1, 0, 1 when x is respectively negative, zero or positive.

This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral.

This gives the following formulas (where a ≠ 0), which are valid over any interval where f is continuous (over larger intervals, the constant C must be replaced by a piecewise constant function):

* ∫|(ax+b)n|dx=sgn⁡(ax+b)(ax+b)n+1a(n+1)+C

when n is odd, and n≠−1.
* ∫|tan⁡ax|dx=−1asgn⁡(tan⁡ax)ln⁡(|cos⁡ax|)+C

when ax∈(nπ−π2,nπ+π2) for some integer n.
* ∫|csc⁡ax|dx=−1asgn⁡(csc⁡ax)ln⁡(|csc⁡ax+cot⁡ax|)+C

when ax∈(nπ,nπ+π) for some integer n.
* ∫|sec⁡ax|dx=1asgn⁡(sec⁡ax)ln⁡(|sec⁡ax+tan⁡ax|)+C

when ax∈(nπ−π2,nπ+π2) for some integer n.
* ∫|cot⁡ax|dx=1asgn⁡(cot⁡ax)ln⁡(|sin⁡ax|)+C

when ax∈(nπ,nπ+π) for some integer n.

If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0. For having a continuous antiderivative, one has thus to add a well chosen step function. If we also use the fact that the absolute values of sine and cosine are periodic with period π, then we get:

### Special functions

Ci, Si: Trigonometric integrals, Ei: Exponential integral, li: Logarithmic integral function, erf: Error function

* ∫Ci⁡(x)dx=xCi⁡(x)−sin⁡x
* ∫Si⁡(x)dx=xSi⁡(x)+cos⁡x
* ∫Ei⁡(x)dx=xEi⁡(x)−ex
* ∫li⁡(x)dx=xli⁡(x)−Ei⁡(2ln⁡x)
* ∫li⁡(x)xdx=ln⁡xli⁡(x)−x
* ∫erf⁡(x)dx=e−x2π+xerf⁡(x)

## Definite integrals lacking closed-form antiderivatives

See also: List of definite integrals

There are some functions whose antiderivatives cannot be expressed in closed form. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below.

* ∫0∞xe−xdx=12π (see also Gamma function)
* ∫0∞e−ax2dx=12πa for a > 0 (the Gaussian integral)
* ∫0∞x2e−ax2dx=14πa3 for a > 0
* ∫0∞x2ne−ax2dx=2n−12a∫0∞x2(n−1)e−ax2dx=(2n−1)!!2n+1πa2n+1=(2n)!n!22n+1πa2n+1

for a > 0, n is a positive integer and !! is the double factorial.
* ∫0∞x3e−ax2dx=12a2 when a > 0
* ∫0∞x2n+1e−ax2dx=na∫0∞x2n−1e−ax2dx=n!2an+1

for a > 0, n = 0, 1, 2, ....
* ∫0∞xne−axbdx=1b a−n+1bΓ(n+1b)
* ∫0∞xex−1dx=π26 (see also Bernoulli number)
* ∫0∞x2ex−1dx=2ζ(3)≈2.40
* ∫0∞x3ex−1dx=π415 (used in the derivation of Planck's law in physics)
* ∫0∞xnex−1dx=Γ(n+1)ζ(n+1) for n>0 (see also Riemann zeta function)
* ∫0∞sin⁡xxdx=π2 (see sinc function and the Dirichlet integral)
* ∫0∞sin2xx2dx=π2
* ∫0π2sinnxdx=∫0π2cosnxdx=(n−1)!!n!!×{1if n is oddπ2if n is even.

(if n is a positive integer and !! is the double factorial).
* ∫−ππcos⁡(αx)cosn(βx)dx={2π2n(nm)|α|=|β(2m−n)|0otherwise

(for α, β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient)
* ∫−ttsinm(αx)cosn(βx)dx=0

(for α, β real, n a non-negative integer, and m an odd, positive integer; since the integrand is odd)
* ∫−ππsin⁡(αx)sinn(βx)dx={(−1)(n+12)(−1)m2π2n(nm)n odd, α=β(2m−n)0otherwise

(for α, β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient)
* ∫−ππcos⁡(αx)sinn(βx)dx={(−1)(n2)(−1)m2π2n(nm)n even, |α|=|β(2m−n)|0otherwise

(for α, β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient)
* ∫−∞∞e−(ax2+bx+c)dx=πaexp⁡[b2−4ac4a]

(where exp[u] is the exponential function eu, and a > 0.)
* ∫0∞xz−1e−xdx=Γ(z)

(where Γ(z) is the Gamma function)
* ∫01(ln⁡1x)pdx=Γ(p+1)
* ∫01xα−1(1−x)β−1dx=Γ(α)Γ(β)Γ(α+β)

(for Re(α) > 0 and Re(β) > 0, see Beta function)
* ∫02πexcos⁡θdθ=2πI0(x) (where I0(x) is the modified Bessel function of the first kind)
* ∫02πexcos⁡θ+ysin⁡θdθ=2πI0(x2+y2)
* ∫−∞∞(1+x2ν)−ν+12dx=νπ Γ(ν2)Γ(ν+12)

(for ν > 0 , this is related to the probability density function of Student's t-distribution)

If the function f has bounded variation on the interval [a,b], then the method of exhaustion provides a formula for the integral: ∫abf(x)dx=(b−a)∑n=1∞∑m=12n−1(−1)m+12−nf(a+m(b−a)2−n).

The "sophomore's dream": ∫01x−xdx=∑n=1∞n−n(=1.29128599706266…)∫01xxdx=−∑n=1∞(−n)−n(=0.78343051071213…) attributed to Johann Bernoulli.

## See also

* Differentiation rules - Rules for computing derivatives of functions
* Incomplete gamma function - Types of special mathematical functions
* Indefinite sum - Inverse of a finite difference
* Integration using Euler's formula - Use of complex numbers to evaluate integrals
* Liouville's theorem (differential algebra)
* List of limits
* List of mathematical identities
* List of mathematical series
* Nonelementary integral - Integrals not expressible in closed-form from elementary functions
* Symbolic integration - In mathematics, computation of an antiderivative in a closed form

## References

1. ↑ Hirsch, Meyer (1810) (in de). Integraltafeln: oder, Sammlung von integralformeln. Duncker & Humblot. https://books.google.com/books?id=8IUAAAAAMAAJ.
2. ↑ Serge Lang . A First Course in Calculus, 5th edition, p. 290
3. ↑ "Reader Survey: log|x| + C", Tom Leinster, The n-category Café, March 19, 2012

## Further reading

* Abramowitz, Milton; Stegun, Irene Ann, eds (1983). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. LCCN 65-12253. ISBN 978-0-486-61272-0.
* (in German) Taschenbuch der Mathematik. 1. Weiß, Jürgen (23 ed.). Thun and Frankfurt am Main: Verlag Harri Deutsch (and B. G. Teubner Verlagsgesellschaft, Leipzig). 1987. ISBN 3-87144-492-8.
* (in English) Table of Integrals, Series, and Products (8 ed.). Academic Press, Inc.. 2015. ISBN 978-0-12-384933-5. (Several previous editions as well.)
* (in English) Integrals and Series. 1–5 (1 ed.). (Nauka) Gordon & Breach Science Publishers/CRC Press. 1988–1992. ISBN 2-88124-097-6. . Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
* Yuri A. Brychkov (Ю. А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 1-58488-956-X / 9781584889564.
* Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. (Many earlier editions as well.)
* Meyer Hirsch [de; de; Meier Hirsch], Integraltafeln oder Sammlung von Integralformeln (Duncker und Humblot, Berlin, 1810)
* Meyer Hirsch [de; de; Meier Hirsch], Integral Tables Or A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln]
* David Bierens de Haan, Nouvelles Tables d'Intégrales définies (Engels, Leiden, 1862)
* Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899)

## External links

### Tables of integrals

* Paul's Online Math Notes
* A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): Indefinite Integrals Definite Integrals
* Math Major: A Table of Integrals
* O'Brien, Francis J. Jr.. "500 Integrals of Elementary and Special Functions". https://www.scribd.com/document/576062422/500-Integrals-of-Elementary-and-Special-Functions. Derived integrals of exponential, logarithmic functions and special functions.
* Rule-based Integration Precisely defined indefinite integration rules covering a wide class of integrands
* Mathar, Richard J. (2012). "Yet another table of integrals". arXiv:1207.5845 [math.CA].

### Derivations

* Victor Hugo Moll, The Integrals in Gradshteyn and Ryzhik

### Online service

* Integration examples for Wolfram Alpha

### Open source programs

* wxmaxima gui for Symbolic and numeric resolution of many mathematical problems

### Videos

* The Single Most Overpowered Integration Technique in Existence. YouTube Video by Flammable Maths on symmetries

|  This article includes a mathematics-related list of lists.

|

* v
* t
* e

Lists of integrals

* Rational functions
* Irrational functions
* Trigonometric functions
* Inverse trigonometric functions
* Hyperbolic functions
* Inverse hyperbolic functions
* Exponential functions
* Logarithmic functions
* Gaussian functions

* v
* t
* e

Calculus

Precalculus|

* Binomial theorem
* Concave function
* Continuous function
* Factorial
* Finite difference
* Free variables and bound variables
* Fundamental theorem
* Graph of a function
* Linear function
* Mean value theorem
* Radian
* Rolle's theorem
* Secant
* Slope
* Tangent

Limits|

* Indeterminate form
* Limit of a function
* One-sided limit
* Limit of a sequence
* Orders of approximation
* (ε, δ)-definition of limit

Differential calculus|

* Chain rule
* Derivative
* Differential
* Differential equation
* Differential operator
* Implicit differentiation
* Inverse functions and differentiation
* L'Hôpital's rule
* Leibniz's rule
* Logarithmic derivative
* Mean value theorem
* Newton's method
* Notation
* Leibniz's notation
* Newton's notation
* Product rule
* Quotient rule
* Regiomontanus' angle maximization problem
* Related rates
* Simplest rules
* Constant factor rule in differentiation
* Linearity of differentiation
* Power rule
* Sum rule in differentiation
* Stationary point
* First derivative test
* Second derivative test
* Extreme value theorem
* Maxima and minima
* Taylor's theorem

Integral calculus|

* Antiderivative
* Arc length
* Constant of integration
* Differentiation under the integral sign
* Fundamental theorem of calculus
* Integral of secant cubed
* Integral of the secant function
* Integration by parts
* Integration by substitution
* Tangent half-angle substitution
* Partial fractions in integration
* Quadratic integral
* Proof that 22/7 exceeds π
* Simplest rules
* Constant factor rule in integration
* Linearity of integration
* Sum rule in integration
* Trapezoidal rule
* Trigonometric substitution

Vector calculus|

* Curl
* Directional derivative
* Divergence
* Divergence theorem
* Gradient
* Gradient theorem
* Green's theorem
* Laplacian
* Stokes' theorem

Multivariable calculus|

* Curvature
* Disc integration
* Divergence theorem
* Exterior
* Gabriel's Horn
* Geometric
* Hessian matrix
* Jacobian matrix and determinant
* Line integral
* Matrix
* Multiple integral
* Partial derivative
* Shell integration
* Surface integral
* Tensor
* Volume integral

Series|

* Abel's test
* Alternating
* Alternating series test
* Arithmetico-geometric sequence
* Binomial
* Cauchy condensation test
* Direct comparison test
* Dirichlet's test
* Euler–Maclaurin formula
* Fourier
* Geometric
* Harmonic
* Infinite
* Integral test for convergence
* Limit comparison test
* Maclaurin
* Power
* Ratio test
* Root test
* Taylor
* Term test

Special functions and numbers|

* Bernoulli numbers
* e (mathematical constant)
* Exponential function
* Natural logarithm
* Stirling's approximation

History of calculus|

* Adequality
* Brook Taylor
* Colin Maclaurin
* Generality of algebra
* Gottfried Wilhelm Leibniz
* Infinitesimal
* Infinitesimal calculus
* Isaac Newton
* Fluxion
* Law of Continuity
* Leonhard Euler
* Method of Fluxions
* The Method of Mechanical Theorems

Lists|

* Differentiation rules
* List of integrals of exponential functions
* List of integrals of hyperbolic functions
* List of integrals of inverse hyperbolic functions
* List of integrals of inverse trigonometric functions
* List of integrals of irrational functions
* List of integrals of logarithmic functions
* List of integrals of rational functions
* List of integrals of trigonometric functions
* List of limits
* List of mathematical symbols
* Lists of integrals

* v
* t
* e

Major topics in Analysis

* Calculus: Integration
* Differentiation
* Differential equations (ordinary \- partial)
* Fundamental theorem of calculus
* Calculus of variations
* Vector calculus
* Tensor calculus
* List of integrals
* Table of derivatives

* Real analysis
* Complex analysis
* Functional analysis
* Fourier analysis
* Harmonic analysis
* Measure theory
* Representation theory

* Functions
* Continuous function
* Special functions
* Limit
* Series
* Infinity

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