[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572875-105":53,"doc-detail-572875-en":126},{"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":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","calculus-2-formula-sheet-chapter-7-u-substitution-chain-rule-and-integration-formulas","Calculus 2 Formula Sheet - Chapter 7 - u-substitution Chain Rule and Integration Formulas","","Calculus 2 formula sheet compiling key results across Chapter 7 and beyond. It includes chain rule derivatives, u-substitution integration templates, and trigonometric and logarithmic forms for common transformations. The sheet further lists integration techniques such as integration by parts, trigonometric substitution cases, and arc length and surface area formulas. It also summarizes parametric curve calculus, polar curve relationships, tangent classification, and area formulas between curves and in polar coordinates.",{"@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/calculus-2-formula-sheet-chapter-7-u-substitution-chain-rule-and-integration-formulas/572875/",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/calculus-2-formula-sheet-chapter-7-u-substitution-chain-rule-and-integration-formulas/572875.png","ImageObject",442,249,{"name":88,"@type":89},"Ezra","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 topics are covered in the Calculus 2 formula sheet?","Question",{"text":108,"@type":109},"It covers chain rule derivatives, u-substitution integration, integration by parts, trigonometric substitution, and formulas for arc length, surface area, and area problems, including parametric and polar curve relationships.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How is u-substitution used in the sheet?",{"text":113,"@type":109},"The sheet provides substitution-based integral templates showing how integrals can be rewritten to match a derivative inside a composition, turning them into simpler forms.",{"name":115,"@type":106,"acceptedAnswer":116},"What are the common trigonometric substitution cases included?",{"text":117,"@type":109},"It lists case-based rules for radicals of forms like sqrt(a^2-b^2) and sqrt(a^2+b^2), assigning specific substitutions and corresponding differentials to rewrite the integral.","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},572875,1792052377,{"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":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":79},1099514068035,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Calculus 2 Formula Sheet  \nChapter 7  \n􀀡􀀡􀀢 (􀀤 ! ) = 􀀤 !  \nChain Rule \"\"! (􀀤 \\# ) = 􀀤 \\# \"\"\\#!  \n' 􀀤 ! 􀀡􀀢 = 􀀤 ! + 􀀩 u-substitution ∫ 􀀤 \\# 􀀡􀀢 = 􀀤 \\# + 􀀩  \n\"\"! + 􀀬􀀭 (􀀢 ). = $! for 􀀢 > 0 Chain Rule \"\"! + 􀀬􀀭 (􀀱 ). = $\\# ⋅ \"\"\\#!\"\"! (􀀬􀀭|􀀢 |) = $! for 􀀢 ≠ 0  \n' 1􀀢 􀀡􀀢 = 􀀬􀀭 (|􀀢 |) + 􀀶 u-substitution ∫ $\\# 􀀡􀀱 = 􀀬􀀭 (|􀀱 |) + 􀀶  \n' tan (􀀢 )􀀡􀀢 = 􀀬􀀭 (|sec (􀀢 ) |) + 􀀶' cot (􀀢 )􀀡􀀢 = 􀀬􀀭 (|sin(􀀢 ) |) + 􀀶  \n\n| 􀀡􀀡􀀢 +􀀬􀁀􀁁%(􀀢 ). = ln~~ ~~(1􀁃)~~ ~~􀀢 Chain Rule \"\"! +􀀬􀁀􀁁%(􀀱 ). = &'~~ ~~($%)! ⋅ \"\"\\#!\u003Cbr>􀀡􀀡􀀢 (􀁃 ! ) = ln (􀁃)􀁃 ! |  |  |  |\n| --- | --- | --- | --- |\n| Chain Rule \"\"!\u003Cbr>' 􀁃 ! 􀀡􀀢\u003Cbr>u-substitution | (􀁃\\# ) = ln (􀁃)􀁃 \\# ⋅ \"\"\\#! |  |  |\n|  | = | 􀁃 !\u003Cbr>ln (􀁃) | + 􀀶 |\n|  | ∫ 􀁃\\# 􀀡􀀢 = &'%(!%) + 􀀶 |  |  |\n\n􀀡􀀡􀀢 +􀁄􀁅􀀭 + $ (􀀢 ). = √1~~ ~~~~ ~~􀀢~~ ,~~​  \nChain Rule \"\"! +􀁄􀁅􀀭 + $ (􀀱 ). = √$~~ ~~\\#\" ⋅ \"\"\\#! 1  \n' ~~  ~~ 􀀡􀀢 = 􀁄􀁅􀀭 + $ (􀀢 ) + 􀀶  \n√1 − 􀀢 ,  \nu-substitution ∫ √$~~ ~~\\#\" 􀀡􀀢 = 􀁄􀁅􀀭 + $ (􀀱) + 􀀶  \n􀀡􀀡􀀢 + 􀀩􀁀􀁄 + $ (􀀢 ). = − √1~~ ~~~~ ~~􀀢~~ ,~~​  \nChain Rule \"\"! + 􀀩􀁀􀁄 + $ (􀀱 ). = − √$~~ ~~\\#\" ⋅ \"\"\\#!  \n􀀡􀀡􀀢 + 􀁈􀁉􀀭 + $ (􀀢 ). = 1~~ ~~􀀢~~ , ~~Chain Rule \"\"! + 􀁈􀁉􀀭 + $ (􀀱). = $~~ ~~$.\\#\" ⋅ \"\"\\#!  \n' ~~ ~~1~~ ~~􀀢~~ ,~~​ 􀀡􀀢 = 􀁈􀁉􀀭 + $ (􀀢 ) + 􀀶  \nu-substitution ∫ $~~ ~~$.\\#\" 􀀡􀀢 = 􀁈􀁉􀀭 + $ (􀀱 ) + 􀀶  \n􀁉~~ , ~~~~ ~~􀀢~~ ,~~​ 􀀡􀀢 = 1􀁉 􀁈􀁉􀀭 + $ J􀀢􀁉K + 􀀶  \n'  \n' √􀁉~~ , ~~~~ ~~􀀢~~ ,~~​ 􀀡􀀢 = 􀁄􀁅􀀭 + $ J􀀢􀁉K + 􀀶  \n􀀡􀀡􀀢 +􀁄􀀤􀀩 + $ (􀀢 ). = 􀀢~~ ~~√􀀢~~ ~~1~~, ~~−~~ ~~1  \nChain Rule \"\"! +􀁄􀀤􀀩 + $ (􀀱 ). = \\#√$\\#\"~~ + ~~$ ⋅ \"\"\\#! 1  \n' ~~  ~~ 􀀡􀀢 = 􀁄􀀤􀀩 + $ (􀀢 ) + 􀀶  \n􀀢 √􀀢 , − 1  \nu-substitution ∫ \\#√$\\#\"~~ + ~~$ 􀀡􀀢 = 􀁄􀀤􀀩 + $ (􀀱 ) + 􀀶  \nChapter 8  \nIntegration By Parts  \n' 􀀱􀀡􀁌 = 􀀱􀁌 − ' 􀁌􀀡􀀱  \n' sec (􀀱 ) 􀀡􀀱 = 􀀬􀀭|sec (􀀱 ) + tan (􀀱 ) | + 􀀩' csc (􀀱 ) 􀀡􀀱 = −􀀬􀀭|csc (􀀱 ) + cot(􀀱 ) | + 􀀩' tan (􀀱 ) 􀀡􀀱 = 􀀬􀀭|􀁄􀀤􀀩 (􀀱 ) | + 􀀩' 􀀩􀁀􀁈 (􀀱 ) 􀀡􀀱 = 􀀬􀀭|􀁄􀁅􀀭 (􀀱) | + 􀀩  \n' 􀁄􀀤􀀩 / (􀀱 )􀀡􀀱 = 12 [sec (􀀱) tan (􀀱 ) + 􀀬􀀭|sec (􀀱 ) + tan (􀀱 ) |] + 􀀩  \nTrigonometric Substitution  \nCase 1: √􀁐 􀀰 − 􀁑 􀀰 where 􀁐 > 􀁒  \n􀀢 = 􀁉􀁄􀁅􀀭 (􀁓)  \n􀀡􀀢 = 􀁉􀀩􀁀􀁄 (􀁓)  \nT􀁉 , − 􀀢 , = 􀁉􀀩􀁀􀁄 (􀁓)  \nCase 2: √􀁐 􀀰 + 􀁑 􀀰 where 􀁐 > 􀁒  \n􀀢 = 􀁉􀁈􀁉􀀭 (􀁓)  \n􀀡􀀢 = 􀁉􀁄􀀤􀀩 ,(􀁓)  \nT􀁉 , − 􀀢 , = 􀁉􀁄􀀤􀀩 (􀁓)  \nCase 3: √􀁑 􀀰 − 􀁐 􀀰 where 􀁐 > 􀁒  \n􀀢 = 􀁉􀁄􀀤􀀩 (􀁓)  \n􀀡􀀢 = 􀁉􀁄􀀤􀀩 (􀁓)􀁈􀁉􀀭 (􀁓)  \nT􀀢 , − 􀁉 , = 􀁉􀁈􀁉􀀭 (􀁓)  \nChapter 9  \nArc Length Formulas  \n􀀡 = ∫ $1~~ ~~+~~ ~~[~~ ~~􀀩for~~´~~􀀡(􀀬=)]􀀣~~!~~​(􀀥􀀬ovoerr􀀡􀀧 01 + 1$$%&2! 􀀯􀀬  \n􀀡 = ∫'$ $1~~ ~~+~~ ~~[~~ ~~􀀳~~´ ~~(􀀴)]~~!~~​ 􀀯􀀴 or 􀀡 = ∫'$ 01 + 1$$&%2! 􀀯􀀴 for 􀀥 = 􀀪 (􀀡) over 􀀫 ≤ 􀀡 ≤ 􀀬  \nSurface Area Formulas  \n􀀵􀀶 = 7 􀀸􀀹􀀺􀀻􀀼  \nRevolve about the x-axis  \n􀁕􀁖 = ∫ 􀁗􀁘􀁙 􀁚􀁛 where 􀀡􀁄 = \\ 1 + ]~~􀀨~~􀀨􀀩􀀪^ 2 􀀯􀀬 􀀽􀀾 􀀡􀁄 = _ 1 + ]~~􀀨~~􀀨􀀪􀀩^ 2 􀀯􀀴  \nRevolve about the y-axis  \n􀀵􀀶 = ∫ 􀀸􀀹􀀺􀀿􀀻􀀼 where 􀀡􀁄 = \\ 1 + ]~~􀀨~~􀀨􀀩􀀪^ 2 􀀯􀀬 􀀽􀀾 􀀡􀁄 = _ 1 + ]~~􀀨~~􀀨􀀪􀀩^2 􀀯􀀴  \nParametric Equation Calculus Formulas  \n􀀢 = 􀁠 (􀁈)  \n􀁡 = 􀁁 (􀁈)  \n􀁈 ∈ 􀁣  \nDerivative  \n!!\"\\# = ~~   ~~\"~~\\#~~$~~ ~~ assuming $$&, ≠ 0 !\\#  \nSecond Derivative  \n􀀯 !􀀴   ~~ ~~􀀯􀀯􀁂 C 􀀯􀀯􀀴􀀬D  \n=  \n􀀯􀀬 !  􀀯􀀬􀀯􀁂  \nArea  \n􀁅 = 7&& \" 􀀴􀀯􀀬 = 7,, /􀀳 (􀁂)􀀩 - (􀁂)􀀯􀁂  \nArc Length  \n􀁆 = 70/ GC~~ ~~􀀯􀀯􀀬􀁂D! + C~~ ~~􀀯􀀯􀀴􀁂D! 􀀯􀁂  \n􀁈 ≤ 􀁂 ≤ 􀁊  \nSurface Area  \n􀁕􀁖 = ' 􀁗􀁘􀁤􀁚􀁛  \nRotate a Parametric Curve C about the x-axis  \n􀁥􀁦 = '􀀲􀀱 2􀁌􀀾_g􀀯􀀯􀀬􀁂h2 + g~~ ~~􀀯􀀯􀀴􀁂h 2 􀀯􀁂 = '􀀲􀀱 2􀁌􀀴_g􀀯􀀯􀀬􀁂h 2 + g~~ ~~􀀯􀀯􀀴􀁂h2 􀀯􀁂  \nRotate a Parametric Curve C about the y-axis  \n􀁥􀁦 = '􀀲􀀱 2􀀾_g􀀯􀀯􀀬􀁂h2 + g~~ ~~􀀯􀀯􀀴􀁂h 2 􀀯􀁂 = '􀀲􀀱 2􀁌􀀬_g􀀯􀀯􀀬􀁂h 2 + g~~ ~~􀀯􀀯􀀴􀁂h 2 􀀯􀁂  \nCalculus of Polar Curves  \n􀀭 = 􀀣 (􀀮)  \n\n| $%\u003Cbr>$& | = | !\"\u003Cbr> !\\# \u003Cbr>!$\u003Cbr>\u003Cbr>!\\# | = | !\"\u003Cbr>!%\u003Cbr>!$\u003Cbr>\u003Cbr>!% | = | !!% 34567(9);\u003Cbr>!!% 34'\u003C5 (9); | = | !!&%567 (9)=4'\u003C5 (9)\u003Cbr>!!&%'\u003C5 (9)>4567 (9) | if | $$&9 ≠ 0 |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n\nHorizontal Tangents  \nPoints on the curve in which $$%9 = 0, but $$&9 ≠ 0  \nVertical Tangents  \nPoints on the curve in which $$&9 = 0, but $$%9 ≠ 0  \nTangent Line at the Pole  \n\"3  $4  \nWe let 􀀭 = 0 and ~~ ~~ = 􀁈􀁉􀀭 (􀁓) if ≠ 0 \"! $9  \nArea  \n􀁩 = 􀁠 (􀁓)  \n􀁪 ≤ 􀁓 ≤ 􀁬  \n􀁦 = ∫ $,[􀁠(􀁓)], 􀀡􀁓 4 1  \n􀁦 = ' 2 􀁩 , 􀀡􀁓 5  \nArea Between Two Curves  \n􀁩 = 􀁠 (􀁓) 􀁉􀀭􀀡 􀁩 = 􀁁 (􀁓)  \n􀁪 ≤ 􀁓 ≤ 􀁬  \n1 4  \n􀁦 = 2 ' (􀁩, 6 − 􀁩,%) 􀀡􀁓 5  \nArc Length  \n􀁩 = 􀁠 (􀁓)  \n􀁪 ≤ 􀁓 ≤ 􀁬  \n􀁥 = '54 _m􀀬􀀬􀀭􀀮n2 + 􀀭2 􀀬􀀮","cbCainzweul61F0v","https://ap.wps.com/l/cbCainzweul61F0v","pdf",194857,10,"English","# Chapter 7\n## Chain Rule and u-substitution\n## Integration by Parts\n## Trigonometric Substitution Cases\n## Arc Length and Surface Area\n# Derivatives and Parametric/POLAR Formulas\n## Derivative and Second Derivative\n## Area and Arc Length\n## Parametric and Polar Curve Calculus","[{\"question\":\"What topics are covered in the Calculus 2 formula sheet?\",\"answer\":\"It covers chain rule derivatives, u-substitution integration, integration by parts, trigonometric substitution, and formulas for arc length, surface area, and area problems, including parametric and polar curve relationships.\"},{\"question\":\"How is u-substitution used in the sheet?\",\"answer\":\"The sheet provides substitution-based integral templates showing how integrals can be rewritten to match a derivative inside a composition, turning them into simpler forms.\"},{\"question\":\"What are the common trigonometric substitution cases included?\",\"answer\":\"It lists case-based rules for radicals of forms like sqrt(a^2-b^2) and sqrt(a^2+b^2), assigning specific substitutions and corresponding differentials to rewrite the integral.\"}]","Calculus 2 Formula Sheet - Chapter 7 - u-substitution Chain Rule and Integration Formulas | PDF",1791929810]