[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572934-105":53,"doc-detail-572934-en":120},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":113,"head_meta":115,"extra_data":117,"updated_unix":119},105,"en","integral-calculus-formula-sheet-derivative-rules-integration-rules-applications","Integral Calculus Formula Sheet - Derivative Rules - Integration Rules - Applications","","Integral Calculus Formula Sheet compiles key reference formulas for derivatives, integrals, and common calculus techniques. It summarizes derivative rules, integral properties, standard integration rules, the Fundamental Theorem of Calculus, and Riemann sums with right, left, and midpoint endpoint approximations. It also includes net change concepts, trigonometric identities, geometry area/volume formulas, physical application relationships such as mass and work, and guidance for integration by parts plus trig integrals.",{"@graph":63,"@context":112},[64,80,95],{"@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":36,"@type":70,"position":76},"https://docshare.wps.com/template/forms/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/integral-calculus-formula-sheet-derivative-rules-integration-rules-applications/572934/",4,{"url":78,"name":59,"@type":81,"author":82,"headline":59,"publisher":85,"fileFormat":88,"inLanguage":57,"description":61,"dateModified":89,"datePublished":89,"encodingFormat":88,"isAccessibleForFree":90,"interactionStatistic":91},"DigitalDocument",{"name":83,"@type":84},"McQueen","Person",{"url":68,"name":86,"@type":87},"DocShare","Organization","application/pdf","2026-10-13",true,{"@type":92,"interactionType":93,"userInteractionCount":4},"InteractionCounter",{"@type":94},"ViewAction",{"@type":96,"mainEntity":97},"FAQPage",[98,104,108],{"name":99,"@type":100,"acceptedAnswer":101},"What derivative rules are provided on the sheet?","Question",{"text":102,"@type":103},"The sheet lists standard derivative rules in a tabular “Derivative Rules” section, covering basic power and common function patterns used in calculus.","Answer",{"name":105,"@type":100,"acceptedAnswer":106},"How does the sheet summarize integration rules and the Fundamental Theorem of Calculus?",{"text":107,"@type":103},"“Integration Rules” gathers common antiderivative formulas, while “Fundamental Theorem of Calculus” states relationships between integrals and antiderivatives over an interval.",{"name":109,"@type":100,"acceptedAnswer":110},"Which numerical approximation methods are included for definite integrals?",{"text":111,"@type":103},"“Riemann Sums” presents endpoint-based and midpoint-based approximations, including Right Endpoint Rule, Left Endpoint Rule, and Midpoint Rule.","https://schema.org",{"og:url":78,"og:type":114,"og:title":59,"og:site_name":86,"og:description":61},"article",{"robots":116,"canonical":78},"index,follow",{"doc_id":118,"site_id":56},572934,1791930159,{"code":4,"msg":5,"data":121},{"doc_id":118,"user_id":122,"nickname":83,"user_avatar":123,"doc_module":9,"category_id":35,"category_name":36,"doc_title":59,"doc_description":61,"doc_content":124,"file_id":125,"file_url":126,"file_type":127,"file_size":128,"view_count":4,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":79,"language":129,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":130,"faqs":131,"seo_title":132,"seo_description":61,"update_tm":119,"read_time":73},5909890329169,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Integral Calculus Formula Sheet  \nDerivative Rules:  \n\n|  􀢊 􀢊􀢞 |  |  | (􀢉) = 􀫙 | 􀝀􀝀􀝔 (􀝔 􀯡 ) = 􀝊􀝔􀯡−1 |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  􀝀 􀝀􀝔 |  | (􀝏􀝅􀝊 􀝔) = 􀜿􀝋􀝏 􀝔 |  |  􀝀 􀝀􀝔 |  | (􀜿􀝋􀝏 􀝔) = − 􀝏􀝅􀝊 􀝔 |  􀝀 􀝀􀝔 | (􀜽􀯫 ) = 􀜽􀯫 􀝈􀝊 􀜽 |  |\n|  􀝀 􀝀􀝔 | (􀝏􀝁􀜿 􀝔) = 􀝏􀝁􀜿 􀝔 􀝐􀜽􀝊 􀝔 |  |  |  􀝀 􀝀􀝔 | (􀜿􀝏􀜿 􀝔) = − 􀜿􀝏􀜿 􀝔 􀜿􀝋􀝐 􀝔 |  |  |  |  |\n|  |  |  |  |  |  |  |  􀝀 􀝀􀝔 |  | (􀝁 􀯫 ) = 􀝁 􀯫 |\n| 􀝀􀝀􀝔 (􀝐􀜽􀝊 􀝔) = 􀝏􀝁􀜿 2􀝔 |  |  |  | 􀝀􀝀􀝔 (􀜿􀝋􀝐 􀝔) = −􀜿􀝏􀜿 2􀝔 |  |  |  |  |  |\n| 􀝀􀝀􀝔 􀵫􀜿􀝂 (􀝔)􀵯 = 􀜿 􀝀􀝀􀝔 􀵫􀝂 (􀝔)􀵯 |  |  |  | 􀝀􀝀􀝔 􀵫􀝂 (􀝔) ± 􀝃 (􀝔)􀵯 = 􀝀􀝀􀝔 􀵫􀝂 (􀝔)􀵯 ± 􀝀􀝀􀝔 􀵫􀝃 (􀝔)􀵯 |  |  |  |  |  |\n| ′ ′ ′\u003Cbr>(􀝂 ⋅ 􀝃) = 􀝂 ⋅ 􀝃 + 􀝂 ⋅ 􀝃 |  |  |  | ′\u003Cbr>′ ′\u003Cbr>􀵬􀝂􀝃􀵰 = 􀝂~~ ~~􀝃2~~ ~~􀝂􀝃~~ ~~ |  |  | 􀢊􀢊􀢞 􁉀􀢌􀵫􀢍 (􀢞)􀵯􁉁 = 􀢌 ′ 􀵫􀢍 (􀢞)􀵯 􀢍 ′(􀢞) |  |  |\n\nProperties of Integrals:  \n\n| 􀶱 􀝇􀝂 (􀝑)􀝀􀝑 = 􀝇 􀶱 􀝂 (􀝑)􀝀􀝑 | 􀶱 [􀝂 (􀝑) ± 􀝃 (􀝑)] 􀝀􀝑 = 􀶱 􀝂 (􀝑)􀝀􀝑 ± 􀶱 􀝃 (􀝑)􀝀􀝑 |\n| --- | --- |\n| 􀶱􀯔􀯔􀝂 (􀝔)􀝀􀝔 = 0 | 􀶱􀯔􀯕􀝂 (􀝔)􀝀􀝔 = − 􀶱􀯕􀯔􀝂 (􀝔)􀝀􀝔 |\n| 􀶱􀯔􀯖􀝂 (􀝔)􀝀􀝔 = 􀶱􀯔􀯕􀝂 (􀝔)􀝀􀝔 + 􀶱􀯕􀯖􀝂 (􀝔)􀝀􀝔 | 􀝂􀯔􀯩􀯘 = 􀜾~~ ~~~~ ~~􀜽 􀶱􀯔􀯕􀝂 (􀝔)􀝀􀝔 |\n| ∫􀯔 􀝂 (􀝔)􀝀􀝔 = 2 ∫0􀯔 􀝂 (􀝔)􀝀􀝔 if f(x) is even | ∫􀯔 􀝂 (􀝔)􀝀􀝔 = 0 if f(x) is odd |\n| 􀶱􀯔􀯕 􀝃 (􀝂 (􀝔))􀝂 ʹ (􀝔)􀝀􀝔 = 􀶱􀯙􀯙(􀯔(􀯕))􀝃 (􀝑)􀝀􀝑 | 􀶱 􀝑􀝀􀝒 = 􀝑􀝒 − 􀶱 􀝒􀝀􀝑 |\n\nIntegration Rules:  \n\n| 􀶱 􀝀􀝑 = 􀝑 + 􀜥\u003Cbr>􀶱 􀝑􀯡 􀝀􀝑 = 􀝑􀝊 + 􀜥􀶱 􀝀􀝑􀝑 = 􀝈􀝊 |􀝑 | + 􀜥\u003Cbr>􀶱 􀝁 􀯨 􀝀􀝑 = 􀝁 􀯨 + 􀜥\u003Cbr>􀶱 􀜽􀯨 􀝀􀝑 = 􀝈1􀝊􀜽 􀜽􀯨 + 􀜥 | 􀶱 􀝏􀝅􀝊 􀝑 􀝀􀝑 = − 􀜿􀝋􀝏 􀝑 + 􀜥\u003Cbr>􀶱 􀜿􀝋􀝏 􀝑 􀝀􀝑 = 􀝏􀝅􀝊 􀝑 + 􀜥\u003Cbr>􀶱 􀝏􀝁􀜿2 􀝑 􀝀u = 􀝐􀜽􀝊 􀝑 + 􀜥\u003Cbr>􀶱 􀜿􀝏􀜿2 􀝑 = − 􀜿􀝋􀝐 􀝑 + 􀜥\u003Cbr>􀶱 􀜿􀝏􀜿 􀝑 􀜿􀝋􀝐 􀝑 􀝀􀝑 = − 􀜿􀝏􀜿 􀝑 + 􀜥\u003Cbr>􀶱 􀝏􀝁􀜿 􀝑 􀝐􀜽􀝊 􀝑 􀝀􀝑 = 􀝏􀝁􀜿 􀝑 + 􀜥 | 􀶱 􀜽~~ ~~2~~ ~~􀝑~~ ~~2 = 1􀜽 􀜽􀝎􀜿􀝐􀜽􀝊 􁉀􀝑􀜽􁉁 + C ∫ 􀶥􀯔􀯗22 = 􀜽􀝎􀜿􀝏􀝅􀝊 􁉀􀯨􀯔􁉁 + 􀜥\u003Cbr>􀶱 􀝑√􀝑􀝀2􀝑−~~ ~~􀜽~~ ~~2 = 1􀜽 􀜽􀝎􀜿􀝏􀝁􀜿 􁉆~~ ~~|􀝑􀜽~~ ~~|􁉇 + 􀜥 |\n| --- | --- | --- |\n\nFundamental Theorem of Calculus:  \n\n| 􀜨 ʹ(􀝔 ) = 􀯗􀯗􀯫 ∫􀯫􀯔 􀝂 (􀝐)􀝀􀝐 = 􀝂 (􀝔 ) where 􀝂(􀝐) is a continuous function on [a, x] . |\n| --- |\n| ∫􀯕􀯔 􀝂 (􀝔 )􀝀􀝔 = 􀜨 (􀜾) − 􀜨 (􀜽), where F(x) is any antiderivative of f(x) . |\n\nRiemann Sums:  \n\n| 􀯡 􀯡\u003Cbr>􀷍 􀜿􀜽􀯜 = 􀜿 􀷍 􀜽􀯜􀯜=1 􀯜=1\u003Cbr>􀯡 􀯡 􀯡 |  |  | 􀯡\u003Cbr>􀶱􀯔􀯕􀝂 (􀝔)􀝀􀝔 = 􀝈􀯡 􀷍 􀝂 (􀜽 + 􀝅􀟂􀝔)􀟂􀝔\u003Cbr>􀯜=1\u003Cbr>􀟂􀝔 = 􀜾~~ ~~~~ ~~􀜽 |\n| --- | --- | --- | --- |\n| 􀷍 􀜽􀯜 + 􀜾􀯜􀯜 = 1 | = 􀷍 􀜽􀯜􀯜 = 1 | + 􀷍 􀜾􀯜􀯜 = 1 |  |\n| 􀯡\u003Cbr>􀷍 1 = 􀝊􀯜=1\u003Cbr>􀯡\u003Cbr>􀷍 􀝅 = 􀝊~~ ~~(􀝊~~ ~~􀯜=1\u003Cbr>􀯡\u003Cbr>􀷍 􀝅 2\u003Cbr>=\u003Cbr>􀝊 (􀝊 + 1)(2􀝊 + 1) 6\u003Cbr>􀯜=1 􀯡\u003Cbr>􀷍 􀝅 3 = 􀵤􀝊~~ ~~(􀝊~~ ~~􀵨􀯜=1\u003Cbr>2 |  |  | 􀷍 (height of 􀝅th rectangle) ⋅ (width of 􀝅th rectangle)􀯜\u003Cbr>Right Endpoint Rule:\u003Cbr>􀯡 􀯡\u003Cbr>􀷍 􀝂 (􀜽 + 􀝅􀟂􀝔) (􀟂􀝔) = 􀷍 ((􀜾~~ ~~~~ ~~􀜽))􀝂 (􀜽 + 􀝅 (􀜾~~ ~~~~ ~~􀜽))􀯜=1 􀯜=1\u003Cbr>Left Endpoint Rule:\u003Cbr>􀯡 􀯡\u003Cbr>􀷍 􀝂 (􀜽 + (􀝅 − 1)􀟂􀝔) (􀟂􀝔) = 􀷍 ((􀜾~~ ~~~~ ~~􀜽))􀝂 (􀜽 + (􀝅 − 1) ~~ ~~(􀜾~~ ~~~~ ~~􀜽))􀯜=1 􀯜=1\u003Cbr>Midpoint Rule:\u003Cbr>􀯡 􀯡\u003Cbr>􀷍 􀝂 (􀜽 + 􀵬 ~~ ~~(􀝅~~ ~~−~~ ~~12)~~ ~~+~~ ~~􀝅􀵰 􀟂􀝔) (􀟂􀝔) = 􀷍 ((􀜾~~ ~~~~ ~~􀜽))􀝂 (􀜽 + 􀵬 ~~ ~~(􀝅~~ ~~−~~ ~~12)~~ ~~+~~ ~~􀝅􀵰 (􀜾~~ ~~~~ ~~􀜽))􀯜 = 1 􀯜 = 1 |\n\nNet Change:  \n\n| Displacement:∫􀯕􀯔 􀝒(􀝔)􀝀􀝔 | Distance Traveled:∫􀯕􀯔 |􀝒(􀝔) |􀝀􀝔 | 􀝏(􀝐) = 􀝏(0) + 􀶱0􀯧 􀝒(􀝔)􀝀􀝔 | 􀜳 (􀝐) = 􀜳(0) + 􀶱0􀯧 􀜳 ʹ (􀝔)􀝀􀝔 |\n| --- | --- | --- | --- |\n\nTrig Formulas:  \n\n| 2 1\u003Cbr>􀝏􀝅􀝊 (x) = 2 (1 − 􀜿􀝋􀝏(2􀝔)) | 􀝐􀜽􀝊 􀝔 = | 􀝏􀝅􀝊 􀝔 \u003Cbr>􀜿􀝋􀝏 􀝔 | 􀝏􀝁􀜿 􀝔 = | 1\u003Cbr>􀜿􀝋􀝏 􀝔 | 􀜿􀝋􀝏(−x) = 􀜿􀝋􀝏(x) | 􀝏􀝅􀝊2 (x) + 􀜿􀝋􀝏2 (x) = 1 |\n| --- | --- | --- | --- | --- | --- | --- |\n| 2 1\u003Cbr>􀜿􀝋􀝏 (x) = 2 (1 + 􀜿􀝋􀝏(2􀝔)) | 􀜿􀝋􀝐 􀝔 = | 􀜿􀝋􀝏 􀝔\u003Cbr>\u003Cbr>􀝏􀝅􀝊 􀝔 | 1\u003Cbr>􀜿􀝏􀜿 􀝔 = ~~ ~~􀝏􀝅􀝊 􀝔 |  | sin (−x) = − sin(x) | 􀝐􀜽􀝊2 (x) + 1 = 􀝏􀝁􀜿2 (x) |\n\nGeometry Fomulas:  \n\n| Area of a Square:\u003Cbr>􀜣 = 􀝏 2 | Area of a Triangle:\u003Cbr>1\u003Cbr>􀜣 = ~~ ~~ 􀜾ℎ 2 | Area of an\u003Cbr>Equilateral Trangle:√3\u003Cbr>􀜣 = ~~ ~~ 􀝏 2\u003Cbr>4 | Area of a Circle:\u003Cbr>􀜣 = 􀟨􀝎2 | Area of a Rectangle:\u003Cbr>􀜣 = 􀜾ℎ |\n| --- | --- | --- | --- | --- |\n\nAreas and Volumes:  \n\n| Area in terms of x (vertical rectangles):􀶱􀯔􀯕 (􀝐􀝋􀝌 − 􀜾􀝋􀝐􀝐􀝋􀝉)􀝀􀝔 | Area in terms of y (horizontal rectangles):\u003Cbr>􀶱􀯖􀯗 (􀝎􀝅􀝃ℎ􀝐 − 􀝈􀝁􀝂􀝐)􀝀􀝕 |\n| --- | --- |\n| General Volumes by Slicing:\u003Cbr>Given: Base and shape of Cross-sections 􀜸 = ∫􀯕􀯔 􀜣 (􀝔)􀝀􀝔 if slices are vertical\u003Cbr>􀜸 = ∫􀯖􀯗 􀜣 (􀝕)􀝀􀝕 if slices are horizontal | Disk Method:\u003Cbr>For volumes of revolution laying on the axis with slices perpendicular to the axis\u003Cbr>􀜸 = ∫􀯕􀯔 􀟨 [􀜴 (􀝔)]2 􀝀􀝔 if slices are vertical\u003Cbr>􀜸 = ∫􀯖􀯗 􀟨 [􀜴 (􀝕)]2 􀝀􀝕 if slices are horizontal |\n| Washer Method:\u003Cbr>For volumes of revolution not laying on the axis with slices perpendicular to the axis\u003Cbr>􀜸 = ∫􀯕􀯔 􀟨 [􀜴 (􀝔)]2 − 􀟨","cbCailGfwFsKXKlg","https://ap.wps.com/l/cbCailGfwFsKXKlg","pdf",319564,"English","# Derivative Rules\n# Properties of Integrals\n# Integration Rules\n# Fundamental Theorem of Calculus\n# Riemann Sums\n## Right Endpoint Rule\n## Left Endpoint Rule\n## Midpoint Rule\n# Net Change\n# Trig Formulas\n# Geometry Formulas\n# Areas and Volumes\n# Physical Applications\n# Integration by Parts","[{\"question\":\"What derivative rules are provided on the sheet?\",\"answer\":\"The sheet lists standard derivative rules in a tabular “Derivative Rules” section, covering basic power and common function patterns used in calculus.\"},{\"question\":\"How does the sheet summarize integration rules and the Fundamental Theorem of Calculus?\",\"answer\":\"“Integration Rules” gathers common antiderivative formulas, while “Fundamental Theorem of Calculus” states relationships between integrals and antiderivatives over an interval.\"},{\"question\":\"Which numerical approximation methods are included for definite integrals?\",\"answer\":\"“Riemann Sums” presents endpoint-based and midpoint-based approximations, including Right Endpoint Rule, Left Endpoint Rule, and Midpoint Rule.\"}]","Integral Calculus Formula Sheet - Derivative Rules - Integration Rules - Applications | PDF"]