List of integrals of inverse trigonometric functions

Trigonometry

Outline History Usage Functions (sin, cos, tan, inverse) Generalized trigonometry
Reference
Identities Exact constants Tables Unit circle
Laws and theorems
Sines Cosines Tangents Cotangents Pythagorean theorem
Calculus
Trigonometric substitution Integrals (inverse functions) Derivatives Trigonometric series
Mathematicians
Hipparchus Ptolemy Brahmagupta al-Hasib al-Battani Regiomontanus Viète de Moivre Euler Fourier
v t e

The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions. For a complete list of integral formulas, see lists of integrals.

The inverse trigonometric functions are also known as the "arc functions".
C is used for the 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.
There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as sin⁻¹, asin, or, as is used on this page, arcsin.
For each inverse trigonometric integration formula below there is a corresponding formula in the list of integrals of inverse hyperbolic functions.

Arcsine function integration formulas

$\int \arcsin(x)\,dx=x\arcsin(x)+{\sqrt {1-x^{2}}}+C$
$\int \arcsin(ax)\,dx=x\arcsin(ax)+{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C$
$\int x\arcsin(ax)\,dx={\frac {x^{2}\arcsin(ax)}{2}}-{\frac {\arcsin(ax)}{4\,a^{2}}}+{\frac {x{\sqrt {1-a^{2}x^{2}}}}{4\,a}}+C$
$\int x^{2}\arcsin(ax)\,dx={\frac {x^{3}\arcsin(ax)}{3}}+{\frac {\left(a^{2}x^{2}+2\right){\sqrt {1-a^{2}x^{2}}}}{9\,a^{3}}}+C$
$\int x^{m}\arcsin(ax)\,dx={\frac {x^{m+1}\arcsin(ax)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {1-a^{2}x^{2}}}}\,dx\,,\quad (m\neq -1)$
$\int \arcsin(ax)^{2}\,dx=-2x+x\arcsin(ax)^{2}+{\frac {2{\sqrt {1-a^{2}x^{2}}}\arcsin(ax)}{a}}+C$
$\int \arcsin(ax)^{n}\,dx=x\arcsin(ax)^{n}\,+\,{\frac {n{\sqrt {1-a^{2}x^{2}}}\arcsin(ax)^{n-1}}{a}}\,-\,n\,(n-1)\int \arcsin(ax)^{n-2}\,dx$
$\int \arcsin(ax)^{n}\,dx={\frac {x\arcsin(ax)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {1-a^{2}x^{2}}}\arcsin(ax)^{n+1}}{a\,(n+1)}}\,-\,{\frac {1}{(n+1)\,(n+2)}}\int \arcsin(ax)^{n+2}\,dx\,,\quad (n\neq -1,-2)$

Arccosine function integration formulas

$\int \arccos(x)\,dx=x\arccos(x)-{\sqrt {1-x^{2}}}+C$
$\int \arccos(ax)\,dx=x\arccos(ax)-{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C$
$\int x\arccos(ax)\,dx={\frac {x^{2}\arccos(ax)}{2}}-{\frac {\arccos(ax)}{4\,a^{2}}}-{\frac {x{\sqrt {1-a^{2}x^{2}}}}{4\,a}}+C$
$\int x^{2}\arccos(ax)\,dx={\frac {x^{3}\arccos(ax)}{3}}-{\frac {\left(a^{2}x^{2}+2\right){\sqrt {1-a^{2}x^{2}}}}{9\,a^{3}}}+C$
$\int x^{m}\arccos(ax)\,dx={\frac {x^{m+1}\arccos(ax)}{m+1}}\,+\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {1-a^{2}x^{2}}}}\,dx\,,\quad (m\neq -1)$
$\int \arccos(ax)^{2}\,dx=-2x+x\arccos(ax)^{2}-{\frac {2{\sqrt {1-a^{2}x^{2}}}\arccos(ax)}{a}}+C$
$\int \arccos(ax)^{n}\,dx=x\arccos(ax)^{n}\,-\,{\frac {n{\sqrt {1-a^{2}x^{2}}}\arccos(ax)^{n-1}}{a}}\,-\,n\,(n-1)\int \arccos(ax)^{n-2}\,dx$
$\int \arccos(ax)^{n}\,dx={\frac {x\arccos(ax)^{n+2}}{(n+1)\,(n+2)}}\,-\,{\frac {{\sqrt {1-a^{2}x^{2}}}\arccos(ax)^{n+1}}{a\,(n+1)}}\,-\,{\frac {1}{(n+1)\,(n+2)}}\int \arccos(ax)^{n+2}\,dx\,,\quad (n\neq -1,-2)$

Arctangent function integration formulas

$\int \arctan(x)\,dx=x\arctan(x)-{\frac {\ln \left(x^{2}+1\right)}{2}}+C$
$\int \arctan(ax)\,dx=x\arctan(ax)-{\frac {\ln \left(a^{2}x^{2}+1\right)}{2\,a}}+C$
$\int x\arctan(ax)\,dx={\frac {x^{2}\arctan(ax)}{2}}+{\frac {\arctan(ax)}{2\,a^{2}}}-{\frac {x}{2\,a}}+C$
$\int x^{2}\arctan(ax)\,dx={\frac {x^{3}\arctan(ax)}{3}}+{\frac {\ln \left(a^{2}x^{2}+1\right)}{6\,a^{3}}}-{\frac {x^{2}}{6\,a}}+C$
$\int x^{m}\arctan(ax)\,dx={\frac {x^{m+1}\arctan(ax)}{m+1}}-{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}x^{2}+1}}\,dx\,,\quad (m\neq -1)$

Arccotangent function integration formulas

$\int \operatorname {arccot}(x)\,dx=x\operatorname {arccot}(x)+{\frac {\ln \left(x^{2}+1\right)}{2}}+C$
$\int \operatorname {arccot}(ax)\,dx=x\operatorname {arccot}(ax)+{\frac {\ln \left(a^{2}x^{2}+1\right)}{2\,a}}+C$
$\int x\operatorname {arccot}(ax)\,dx={\frac {x^{2}\operatorname {arccot}(ax)}{2}}+{\frac {\operatorname {arccot}(ax)}{2\,a^{2}}}+{\frac {x}{2\,a}}+C$
$\int x^{2}\operatorname {arccot}(ax)\,dx={\frac {x^{3}\operatorname {arccot}(ax)}{3}}-{\frac {\ln \left(a^{2}x^{2}+1\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+C$
$\int x^{m}\operatorname {arccot}(ax)\,dx={\frac {x^{m+1}\operatorname {arccot}(ax)}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}x^{2}+1}}\,dx\,,\quad (m\neq -1)$

Arcsecant function integration formulas

$\int \operatorname {arcsec}(x)\,dx=x\operatorname {arcsec}(x)\,-\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}}\right)\,+\,C=x\operatorname {arcsec}(x)-\operatorname {arcosh} |x|+C$
$\int \operatorname {arcsec}(ax)\,dx=x\operatorname {arcsec}(ax)-{\frac {1}{a}}\,\operatorname {arcosh} |ax|+C$
$\int x\operatorname {arcsec}(ax)\,dx={\frac {x^{2}\operatorname {arcsec}(ax)}{2}}-{\frac {x}{2\,a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C$
$\int x^{2}\operatorname {arcsec}(ax)\,dx={\frac {x^{3}\operatorname {arcsec}(ax)}{3}}\,-\,{\frac {\operatorname {arcosh} |ax|}{6\,a^{3}}}\,-\,{\frac {x^{2}}{6\,a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}\,+\,C$
$\int x^{m}\operatorname {arcsec}(ax)\,dx={\frac {x^{m+1}\operatorname {arcsec}(ax)}{m+1}}\,-\,{\frac {1}{a\,(m+1)}}\int {\frac {x^{m-1}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}}\,dx\,,\quad (m\neq -1)$

Arccosecant function integration formulas

$\int \operatorname {arccsc}(x)\,dx=x\operatorname {arccsc}(x)\,+\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}}\right)\,+\,C=x\operatorname {arccsc}(x)\,+\,\operatorname {arcosh} |x|\,+\,C$
$\int \operatorname {arccsc}(ax)\,dx=x\operatorname {arccsc}(ax)+{\frac {1}{a}}\,\operatorname {artanh} \,{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C$
$\int x\operatorname {arccsc}(ax)\,dx={\frac {x^{2}\operatorname {arccsc}(ax)}{2}}+{\frac {x}{2\,a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C$
$\int x^{2}\operatorname {arccsc}(ax)\,dx={\frac {x^{3}\operatorname {arccsc}(ax)}{3}}\,+\,{\frac {1}{6\,a^{3}}}\,\operatorname {artanh} \,{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}\,+\,{\frac {x^{2}}{6\,a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}\,+\,C$
$\int x^{m}\operatorname {arccsc}(ax)\,dx={\frac {x^{m+1}\operatorname {arccsc}(ax)}{m+1}}\,+\,{\frac {1}{a\,(m+1)}}\int {\frac {x^{m-1}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}}\,dx\,,\quad (m\neq -1)$

See also

Differentiation of trigonometric functions – Mathematical process of finding the derivative of a trigonometric function
List of trigonometric identities
Lists of integrals

References

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 Definite integrals

Read other articles:

Naval Staff (Sweden)

Naval_Staff_(Sweden)

Military unit Naval StaffMarinstabenCoat of arms of the Swedish Naval Staff.Active1907–1994, 2019–presentCountrySwedenAllegianceSwedish Armed ForcesBranchSwedish NavyTypeStaffRoleOperational, territorial and tactical operationsGarrison/HQMusköMarchFlottans paradmarsch (Wagner)[1]CommandersChief of StaffCapt (N) Håkan NilssonMilitary unit Naval Staff[2] (Swedish: Marinstaben, MS) is the staff of the Chief of the Swedish Navy. Established in 1907, it originated from the Fl...

Shepherd Neolithic

Shepherd_Neolithic

Style of ancient industry in Lebanon Shepherd NeolithicGeographical rangeLebanonPeriodEpipaleolithicDatescirca 10,200 B.C.E. — circa 8,800 B.C.E.Type siteQaa, MaqneMajor sitesHermel, Kamouh el Hermel, Douris, Hermel, Kamouh el Hermel, Qalaat Tannour, Rayak North, Riha StationPreceded byNatufian cultureFollowed byPre-Pottery Neolithic A QaaMaqneDourisHermelKamouh el HermelQalaat TannourRayak NorthRiha Stationclass=notpageimage| Map of Lebanon showing important sites that were occupied in the...

Achmad Alf Arslan Djunaid

Achmad_Alf_Arslan_Djunaid

Achmad Alf Arslan Djunaid Wali Kota Pekalongan ke-22Masa jabatan17 Februari 2016 – 7 September 2017PresidenJoko WidodoGubernurGanjar Pranowo Wali Kota Pekalongan|WakilSaelany Machfudz PendahuluMuhammad Basyir Ahmad SyawiePenggantiSaelany Machfudz Wakil Wali Kota Pekalongan ke-3Masa jabatan5 Juli 2010 – 9 Agustus 2015PresidenSusilo Bambang Yudhoyono Joko WidodoGubernurBibit Waluyo Ganjar PranowoWali KotaMuhammad Basyir Ahmad Syawie PendahuluAbu AlmafachirPengganti...

Katedral Wrexham

Katedral_Wrexham

Katedral WrexhamCathedral Church of Our Lady of Sorrows (St Mary's)Katedral WrexhamKoordinat: 53°02′51″N 2°59′55″W / 53.0474°N 2.9986°W / 53.0474; -2.9986LokasiWrexham,NegaraWalesDenominasiGereja Katolik RomaSejarahTanggal konsekrasi1857ArsitekturArsitekE. W. PuginGayaNeo-GotikDibangun1857AdministrasiKeuskupanKeuskupan WrexhamProvinsiCardiffKlerusUskupPeter Brignall Gereja Katedral Bunda dari Kesedihan atau Cathedral Church of Our Lady of Sorrows, yang juga...

Funeral (Lil Wayne album)

Funeral_(Lil_Wayne_album)

2020 studio album by Lil WayneFuneralStudio album by Lil WayneReleasedJanuary 31, 2020Recorded2016–2019GenreHip hop[1]Length76:04Label Young Money Republic Producer Aaron Z Alex Delicata B Ham Ben Billions Benny Wond3r Bijan Amir Blue Cheeze Bobby Keyz Brandon Finessin Charlie Handsome Chill Shump Cool & Dre Dunk Rock Kamo Loctor Duke LostKidSamy Louis Haze Jahlil Beats Javar Rockamore Mannie Fresh Manny Galvez Mike Will Made It MonstaBeatz Murda Beatz Prxz Rex Cudo R!O ...

Clare Gerada

London based general practitioner DameClare GeradaDBE FRCP FRCPsych FRCGPGerada in 2011BornClare Mary Louise Francis GeradaNovember 1959 (age 64)NigeriaCitizenshipBritish and MalteseEducationMB BS, 1983, UCL Medical SchoolOccupationGeneral practitionerKnown forChairperson of RCGPSpouseSimon WesselyChildren2Medical careerProfessionMedical doctorSub-specialtiesMental health (especially of health care workers), substance misuse, and problem gambling Dame Clare Mary Lou...

You (Time Person of the Year)

You_(Time_Person_of_the_Year)

2006 issue of Time magazine For other uses, see You (disambiguation). Cover of the December 25, 2006 issue. Gray area is a reflective mirror surface. You was the official choice for Time's Person of the Year in 2006. The magazine set out to recognize the millions of people who anonymously contribute user-generated content to websites such as YouTube, MySpace, Facebook, Wikipedia and other wikis, and the multitudes of other websites featuring user contribution.[1][2] While...

卡洛斯·梅内姆

此條目需要擴充。 (2015年11月27日)请協助改善这篇條目，更進一步的信息可能會在討論頁或扩充请求中找到。请在擴充條目後將此模板移除。卡洛斯·梅内姆阿根廷總統府官方照片第47任阿根廷總統任期1989年7月8日—1999年12月10日副总统爱德华多·杜阿尔德卡洛斯·鲁考夫（英语：Carlos Ruckauf）前任劳尔·阿方辛个人资料出生(1930-07-02)1930年7月2日阿根廷拉里奥哈省阿尼利亚�...

DJ Snake

French DJ and record producer (born 1986) DJ SnakeDJ Snake in 2020Background informationBirth nameWilliam Sami Étienne GrigahcineBorn (1986-06-13) 13 June 1986 (age 37)Paris, FranceGenres EDM trap moombahton dubstep OccupationsDJrecord producersongwriterYears active2000–presentLabels Geffen Premiere Classe Starsick Columbia Interscope Mad Decent Spinnin' Universal Music Polydor[1] Websitedjsnake.comMusical artist William Sami Étienne Grigahcine (born 13 June 1986), known by h...

2020年夏季奧林匹克運動會波蘭代表團

2020年夏季奧林匹克運動會波蘭代表團

2020年夏季奥林匹克运动会波兰代表團波兰国旗IOC編碼POLNOC波蘭奧林匹克委員會網站olimpijski.pl（英文）（波兰文）2020年夏季奥林匹克运动会（東京）2021年7月23日至8月8日（受2019冠状病毒病疫情影响推迟，但仍保留原定名称）運動員206參賽項目24个大项旗手开幕式：帕维尔·科热尼奥夫斯基（游泳）和马娅·沃什乔夫斯卡（自行车）[1]闭幕式：卡罗利娜·纳亚（皮划艇）&#...

List of United States Supreme Court cases by the Vinson Court

List_of_United_States_Supreme_Court_cases_by_the_Vinson_Court

Supreme Court of the United StatesVinson CourtStone Court ← → Warren CourtJune 24, 1946 – September 8, 1953(7 years, 76 days)SeatSupreme Court BuildingWashington, D.C.No. of positions9Vinson Court decisions This is a partial chronological list of cases decided by the United States Supreme Court during the Vinson Court, the tenure of Chief Justice Frederick Moore Vinson from June 24, 1946 through September 8, 1953. Case name Citation Summary United States v. Carmack ...

2002 NCAA Division I-AA Football Championship Game

2002_NCAA_Division_I-AA_Football_Championship_Game

College football game This article is about the Division I-AA (now FCS) championship game. For the 2002 Division I-A (now FBS) championship game, see 2002 Rose Bowl. College football game2002 NCAA Division I-AA Football ChampionshipI-AA National Championship Game Western Kentucky Hilltoppers McNeese State Cowboys (11–3) (12–1) Gateway Southland 34 14 Head coach: Jack Harbaugh Head coach: Tommy Tate TSNSeed 15— TSNSeed 11 1234 Total Western Kentucky 710710 34 McNeese ...

Business Professionals of America

Business_Professionals_of_America

Career and technical student organization This article may rely excessively on sources too closely associated with the subject, potentially preventing the article from being verifiable and neutral. Please help improve it by replacing them with more appropriate citations to reliable, independent, third-party sources. (September 2023) (Learn how and when to remove this message) Business Professionals of AmericaBPA logoAbbreviationBPAFormation1966; 58 years ago (1966)TypeYouth ...

Angrogna

Angrogna komune di Italia Tempat Negara berdaulatItaliaDaerah di ItaliaPiemonteKota metropolitan di ItaliaKota Metropolitan Turin NegaraItalia PendudukTotal810 (2023 )GeografiLuas wilayah38,88 km² [convert: unit tak dikenal]Ketinggian782 m Berbatasan denganBricherasio Luserna San Giovanni Prali Perrero Prarostino San Germano Chisone Torre Pellice Villar Pellice Pramollo SejarahSanto pelindungLaurensius Informasi tambahanKode pos10060 Zona waktuUTC+1 UTC+2 Kode telepon0121 ID ISTAT...

Vilavancode taluk

Vilavancode_taluk

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources.Find sources: Vilavancode taluk – news · newspapers · books · scholar · JSTOR (March 2024) 8°20′42″N 77°02′31″E / 8.3451075°N 77.0420246°E / 8.3451075; 77.0420246 Vilavancode taluk is a taluk located in Kanyakumari district, Tami...

Capacitive coupling

Capacitive_coupling

Transfer of energy between circuits Coupling capacitor redirects here. Not to be confused with Decoupling capacitor. DC blocking capacitor redirects here. Not to be confused with Bypass capacitor. This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (August 2023) (Learn how and when to remove this message) Capacitive coupling is the transfer of energy within an ele...

Silvester I

Silvester I Påve 314–335NamnSilvesterFöddokäntDöd31 december 335FöreträdareMiltiadesEfterträdareMarkusPåve i 21 år, 11 månader och 0 dagar Sankt Silvester I Sankt Silvester I PåveVördas inomRomersk-katolska kyrkanOrtodoxa kyrkanHelgedomBasilikan Santi Silvestro e Martino ai Monti i Rom Silvester I, född i Rom, var påve från den 31 januari 314 till sin död, den 31 december 335. Silvester I vördas som helgon i Romersk-katolska kyrkan, med minnesdag den 31 december. Biografi S...

Half-mast

Flag flying below the summit of a ship mast, a pole on land, or a pole on a building The Finnish flag flying at half-mast after the 2011 Norway attacks The American flag flying at half-mast in Buchenwald, Thuringia, Nazi Germany, on 19 April 1945 after the death of US President Franklin Roosevelt Half-mast or half-staff (American English) refers to a flag flying below the summit of a ship mast, a pole on land, or a pole on a building. In many countries this is seen as a symbol of respect, mou...

Íþróttafélagið Þór

Íþróttafélagið_Þór

Íþróttafélagið ÞórColori sociali Bianco, rosso Dati societariCittàAkureyri Paese Islanda FederazioneÍSÍ Fondazione1915 Presidente Nói Björnsson Discipline Bowling Calcio Freccette Pallacanestro Pallamano Pugilato Sport elettronici Taekwondo Sito web ufficiale Modifica dati su Wikidata · Manuale L'Íþróttafélagið Þór, meglio noto come Þór Akureyri o solamente Þór, è una società polisportiva islandese con sede ad Akureyri. La società venne fondata il 6 gi...

Katakana

Japanese syllabary For the Unicode block, see Katakana (Unicode block). Katakana片仮名カタカナScript type Syllabary Time period~800 CE to the presentDirectionVertical right-to-left, left-to-right LanguagesJapanese, Ryukyuan, Hachijō, Ainu[1]Taiwanese Hokkien, Palauan (formerly)Related scriptsParent systemsOracle bone scriptSeal scriptClerical scriptRegular script (kanji)Man'yōganaKatakanaSister systemsHiraganaISO 15924ISO 15924Kana (411), KatakanaUnicodeUnic...