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It includes the fundamental theorem of calculus, Riemann sum formulations with endpoint and midpoint rules, and net change concepts linking displacement, distance, and accumulated quantities. The sheet also covers common trigonometric and geometric formulas, area and volume methods (disk, washer, shell), and selected physical application formulas for mass, work, force, and pressure.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/integral-calculus-formula-sheet-derivative-integral-and-application-rules/571590/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/integral-calculus-formula-sheet-derivative-integral-and-application-rules/571590.png","ImageObject",442,249,{"name":88,"@type":89},"Cipher","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What derivative rules are included in the sheet?","Question",{"text":108,"@type":109},"The sheet lists standard derivative rules for polynomial powers and common functions such as sin, cos, tan, sec, csc, and exponents (e.g., ax, ex), along with derivative patterns for composed functions.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Which integration methods are summarized?",{"text":113,"@type":109},"It provides integration rules covering substitution-style integrals, trigonometric integral identities, and a guide for integration by parts identifying u and dv choices based on function types.",{"name":115,"@type":106,"acceptedAnswer":116},"How does the sheet connect integrals to real-world quantities?",{"text":117,"@type":109},"It includes physical applications using integrals for mass with variable density, work with position-dependent force, and force/pressure with depth-dependent quantities, linking these to calculus-based accumulation.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},571590,1792052386,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":79,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":73},687208528416,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Integral Calculus Formula Sheet  \nDerivative Rules:  \n\n| ddx~~ ~~ 􀀋 c 􀀌 􀀠 0 | ~~ ~~ddx~~ ~~ 􀀋 x n 􀀌 􀀠 nx n 􀀐1 |  |\n| --- | --- | --- |\n| ddx~~ ~~ 􀀋sin x􀀌 􀀠 cos xddx~~ ~~ 􀀋 sec x􀀌 􀀠 sec x tan xddx~~ ~~ 􀀋 tan x􀀌 􀀠 sec2 x | ddx~~ ~~ 􀀋 cos x􀀌 􀀠 􀀐 sin xddx~~ ~~ 􀀋 csc x􀀌 􀀠 􀀐 csc x cot xddx~~ ~~ 􀀋 cot x􀀌 􀀠 􀀐 csc2 x | ddx~~ ~~ 􀀋 ax􀀌 􀀠 ax ln a\u003Cbr>ddx~~ ~~ 􀀋 ex􀀌 􀀠 ex |\n| ~~ ~~ddx~~ ~~ 􀀋cf 􀀋 x 􀀌􀀌 􀀠 c ~~ ~~ddx~~ ~~ 􀀋f 􀀋 x 􀀌􀀌 | ~~ ~~ddx~~ ~~ 􀀋f 􀀋 x 􀀌 􀁲 g 􀀋 x 􀀌􀀌 􀀠 ~~ ~~ddx~~ ~~ 􀀋f 􀀋 x 􀀌􀀌 􀁲 ~~ ~~ddx~~ ~~ 􀀋 g 􀀋 x 􀀌􀀌 |  |\n| 􀀋f 􀂘 g 􀀌 􀁣 􀀠 f 􀁣 􀂘 g 􀀎 f 􀂘 g 􀁣 | 􀂧 f 􀂷 􀁣 f g􀁣 􀀐fg􀁣\u003Cbr>􀂨  􀂸 􀀠  2 \u003Cbr>􀂩g 􀂹 g | ~~ ~~ddx~~ ~~ 􀀋 f 􀀋 g 􀀋 x 􀀌􀀌􀀌 􀀠 f 􀁣 􀀋 g 􀀋 x 􀀌􀀌 g 􀁣 􀀋 x 􀀌 |\n\nProperties of Integrals:  \n\n| 􀂳 kf (u)du 􀀠 k 􀂳 f (u)du | 􀂳􀀾f (u) 􀁲 g(u) 􀁀du 􀀠 􀂳 f (u)du 􀁲 􀂳 g (u)du |\n| --- | --- |\n| a\u003Cbr>􀂳 f (x)dx 􀀠 0\u003Cbr>a | b a\u003Cbr>􀂳 f (x)dx 􀀠 􀀐 􀂳 f (x)dx\u003Cbr>a b |\n| c b c\u003Cbr>􀂳 f (x)dx 􀀠 􀂳 f (x)dx 􀀎 􀂳 f (x)dx\u003Cbr>a a b | b\u003Cbr>fave 􀀠 b~~ ~~~~ ~~a~~ ~~ 􀂳 f (x)dx\u003Cbr>a |\n| a a\u003Cbr>􀂳 f (x)dx 􀀠 2􀂳 f (x)dx if f(x) is even\u003Cbr>􀀐a 0 | a\u003Cbr>􀂳 f (x)dx 􀀠 0 if f(x) is odd\u003Cbr>􀀐a |\n| b f (b)\u003Cbr>􀂳 g ( f (x)) f 􀁣 (x)dx 􀀠 􀂳 g (u)du\u003Cbr>a f (a) | 􀂳 udv 􀀠 uv 􀀐 􀂳 vdu |\n\nIntegration Rules:  \n\n| 􀂳 du 􀀠 u 􀀎 C |  | 􀂳 sin u du 􀀠 􀀐 cosu 􀀎 C\u003Cbr>􀂳 cosu du 􀀠 sin u 􀀎 C\u003Cbr>􀂳 sec2 u du 􀀠 tan u 􀀎 C\u003Cbr>􀂳 csc2 u 􀀠 􀀐 cot u 􀀎 C\u003Cbr>􀂳 cscu cot u du 􀀠 􀀐 cscu 􀀎 C\u003Cbr>􀂳 secu tan u du 􀀠 sec u 􀀎 C | 􀂳 ~~ ~~a~~ ~~2du􀀎u~~ ~~2~~ ~~ 􀀠 ~~ ~~1a~~ ~~ arctan 􀂨􀂧􀂩 ~~ ~~ua~~ ~~ 􀂷􀂹􀂸 􀀎 C\u003Cbr>􀂳 ~~ ~~d2~~ ~~ 􀀠 arcsin  ~~ ~~ua~~ ~~ 􀂷􀂹􀂸 􀀎 C\u003Cbr>􀂳 ~~ ~~u~~ ~~du2u􀀐~~ ~~ 􀀠 ~~ ~~1a~~ ~~ arcsec 􀂧􀂩􀂨 ua~~ ~~ 􀂷􀂹􀂸 􀀎 C\u003Cbr>|\n| --- | --- | --- | --- |\n| 􀂳un du\u003Cbr>􀂳 ~~ ~~duu 􀀠 | n􀀎1\u003Cbr>􀀠 un􀀎1 􀀎 C ln u  􀀎 C |  |  |\n| 􀂳eu du 􀀠 eu 􀀎 C\u003Cbr>􀂳au du 􀀠 ln1a~~ ~~ au 􀀎 C |  |  |  |\n\nFundamental Theorem of Calculus:  \n\n| F ' 􀀋 x 􀀌 􀀠 ddx~~ ~~ 􀂳 ax f 􀀋 t 􀀌 dt 􀀠 f 􀀋 x 􀀌 where f 􀀋 t 􀀌 is a continuous function on [a, x] . |\n| --- |\n| 􀂳 ba f 􀀋 x 􀀌 dx 􀀠 F 􀀋 b 􀀌􀀐 F 􀀋 a 􀀌 , where F(x) is any antiderivative of f(x) . |\n\nRiemann Sums:  \n\n|  cai 􀀠 c aii 􀀠 1 i 􀀠 1\u003Cbr> ai 􀀎 bi 􀀠  ai 􀀎  bii 􀀠 1 i 􀀠 1 i 􀀠 1 | b\u003Cbr>􀂳 f\u003Cbr>a | (x)dx 􀀠 limn􀁯􀁦 |  f (a 􀀎 i􀀧x)􀀧xi 􀀠 1 |\n| --- | --- | --- | --- |\n|  | 􀀧x 􀀠 b~~ ~~~~ ~~a |  |  |\n|  1 􀀠 ni 􀀠 1\u003Cbr>􀂦in􀀠1 i 􀀠 n(n2􀀎~~ ~~1)\u003Cbr>􀂦i n􀀠1 i 2 􀀠 n(n􀀎1)6(2n~~ ~~􀀎~~ ~~1)\u003Cbr>􀂦i n􀀠1 i3 􀀠 􀂪􀂬􀂫 n(n2􀀎~~ ~~1)~~ ~~ 􀂺􀂼􀂻 2 | 􀂦􀀋height of ith rectangle 􀀌􀂘 􀀋width of ith rectangle 􀀌\u003Cbr>i\u003Cbr>Right Endpoint Rule:\u003Cbr> f (a 􀀎 i􀀧x)(􀀧x) 􀀠  ((ba)~~ ~~ )f (a 􀀎 i ~~ ~~(ba)~~ ~~ )\u003Cbr>i 􀀠 1 i 􀀠 1\u003Cbr>Left Endpoint Rule:\u003Cbr> f (a 􀀎 (i 􀀐1)􀀧x)(􀀧x) 􀀠  ((ba)~~ ~~ )f (a 􀀎 (i 􀀐1) ~~ ~~(ba)) i 􀀠 1 i 􀀠 1\u003Cbr>Midpoint Rule:\u003Cbr> f (a 􀀎 􀀋 ~~ ~~(i~~ ~~􀀐1)2􀀎i~~ ~~ 􀀌 􀀧x)(􀀧x) 􀀠  ((ba)~~ ~~ )f (a 􀀎 􀀋 ~~ ~~(i~~ ~~􀀐1)2􀀎i~~ ~~ 􀀌 ~~ ~~(ba)~~ ~~ )\u003Cbr>i 􀀠 1 i 􀀠 1 |  |  |\n\nNet Change:  \n\n| b\u003Cbr>Displacement: 􀂳v(x)dx\u003Cbr>a | b\u003Cbr>Distance Traveled: 􀂳 v(x) dx\u003Cbr>a | t\u003Cbr>s(t ) 􀀠 s(0) 􀀎 􀂳v(x)dx\u003Cbr>0 | t\u003Cbr>Q (t ) 􀀠 Q(0) 􀀎 􀂳 Q 􀁣 (x)dx\u003Cbr>0 |\n| --- | --- | --- | --- |\n\nTrig Formulas:  \n\n| sin 2 (x) 􀀠 ~~ ~~12~~ ~~ 􀀋 1􀀐 cos(2x) 􀀌 | tan x 􀀠 | sin x\u003Cbr>cos x | sec x 􀀠 | 1\u003Cbr>cos x | cos(􀀐x) 􀀠 cos(x) | sin 2 (x) 􀀎 cos2 (x) 􀀠 1 |\n| --- | --- | --- | --- | --- | --- | --- |\n| cos2 (x) 􀀠 ~~ ~~12~~ ~~ 􀀋 1􀀎 cos(2x) 􀀌 | cot x 􀀠 | cos x\u003Cbr>sin x | csc x 􀀠 | 1\u003Cbr>sin x | sin(􀀐x) 􀀠 􀀐 sin(x) | tan 2 (x) 􀀎1 􀀠 sec2 (x) |\n\nGeometry Fomulas:  \n\n| Area of a Square: A 􀀠 s 2 | Area of a Triangle: A 􀀠 ~~ ~~12~~ ~~ bh | Area of an\u003Cbr>Equilateral Trangle: A 􀀠  3  s 2\u003Cbr>4\u003Cbr>| Area of a Circle: A 􀀠 􀁓 r2 | Area of a\u003Cbr>Rectangle: A 􀀠 bh |\n| --- | --- | --- | --- | --- |\n\nAreas and Volumes:  \n\n| Area in terms of x (vertical rectangles):\u003Cbr>b\u003Cbr>􀂳 (top 􀀐 bottom)dx\u003Cbr>a |  |  | Area in terms of y (horizontal rectangles): d\u003Cbr>􀂳 (right 􀀐 left)dy\u003Cbr>c |\n| --- | --- | --- | --- |\n| General Volumes by Slicing:\u003Cbr>Given: Base and shape of Cross‐sections |  |  | Disk Method:\u003Cbr>For volumes of revolution laying on the axis with slices perpendicular to the axis\u003Cbr>b\u003Cbr>V 􀀠 􀂳􀁓 􀀾 R (x) 􀁀 2 dx if slices are vertical a\u003Cbr>d\u003Cbr>V 􀀠 􀂳􀁓 􀀾 R (y ) 􀁀 2 dy if slices are horizontal c |\n| b\u003Cbr>V 􀀠 􀂳 A(x)dx\u003Cbr>a | if slices are vertical |  |  |\n| d\u003Cbr>V 􀀠 􀂳","cbCaiq3vt6OtMXoT","https://ap.wps.com/l/cbCaiq3vt6OtMXoT","pdf",162647,"English","# Derivative Rules\n# Properties of Integrals\n# Integration Rules\n# Fundamental Theorem of Calculus\n# Riemann Sums\n# Net Change and Motion Models\n# Trigonometric Formulas\n# Geometry, Areas, and Volumes\n# Physical Applications","[{\"question\":\"What derivative rules are included in the sheet?\",\"answer\":\"The sheet lists standard derivative rules for polynomial powers and common functions such as sin, cos, tan, sec, csc, and exponents (e.g., ax, ex), along with derivative patterns for composed functions.\"},{\"question\":\"Which integration methods are summarized?\",\"answer\":\"It provides integration rules covering substitution-style integrals, trigonometric integral identities, and a guide for integration by parts identifying u and dv choices based on function types.\"},{\"question\":\"How does the sheet connect integrals to real-world quantities?\",\"answer\":\"It includes physical applications using integrals for mass with variable density, work with position-dependent force, and force/pressure with depth-dependent quantities, linking these to calculus-based accumulation.\"}]","Integral Calculus Formula Sheet - Derivative, Integral, and Application Rules | PDF",1791922526]