[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572892-105":53,"doc-detail-572892-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","maths-methods-club-integration-cheat-sheet-sample-anti-differentiation-definite-integrals-rules","Maths-Methods-Club-Integration-Cheat-Sheet-Sample - Anti-Differentiation & Definite Integrals Rules","","Integration notes covering key properties of integrals, including linearity with plus/minus splits and how constants affect antiderivatives. Includes a dedicated section on forgetting the constant, with reminders about adding “+ c” after integrating and how expressions in brackets must be linear for the antidifferentiation rules to apply. Summarizes several anti-differentiation patterns for powers, logarithms, and trigonometric functions, then consolidates the definite integral rule using an antiderivative and fundamental integral properties.",{"@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":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/maths-methods-club-integration-cheat-sheet-sample-anti-differentiation-definite-integrals-rules/572892/",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/maths-methods-club-integration-cheat-sheet-sample-anti-differentiation-definite-integrals-rules/572892.png","ImageObject",442,249,{"name":88,"@type":89},"Sophia Brooks","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-17","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":47},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"Why must the constant of integration be included after integrating?","Question",{"text":108,"@type":109},"The notes highlight that forgetting to add the constant leads to incomplete antiderivatives. Add “+ c” after integrating to capture all possible solutions.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"When do the antidifferentiation rules apply?",{"text":113,"@type":109},"They apply only when the expression inside the brackets is linear. For other kinds, expand if possible before integrating or use a CAS calculator.",{"name":115,"@type":106,"acceptedAnswer":116},"How is a definite integral computed using an antiderivative?",{"text":117,"@type":109},"If the function is continuous on [a, b], the notes state that the definite integral equals the antiderivative evaluated at b minus its value at a, i.e., F(b) − F(a).","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},572892,1792002591,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":47,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"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":4},962084925636,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Integration  \n\n| \u003Cbr>Properties of Integrals\u003Cbr>\u003Cbr>• ∫ 􀝂 ± 􀝃 (􀝔) 􀝀􀝔 = ∫ 􀝂 􀝀􀝔 ± ∫ 􀝃  􀝀􀝔\u003Cbr>• ∫ 􀝇􀝂 􀝀􀝔 = 􀝇 ∫ 􀝂 􀝀􀝔, where 􀝇 is a real number. |  | Forgetting to include the constant |  |\n| --- | --- | --- | --- |\n|  |  | ‘+􀜿’after integrating or\u003Cbr>including‘􀝀􀝔’when\u003Cbr>integrating. | |\n|  |  |  |  |\n|  | Note: The antidifferentiation rules |  |  |\n|  | described ONLY apply if the expression inside the brackets\u003Cbr>is linear. \u003Cbr>If the expression is of any other kind, you may need to expand it if possible before integrating, or you must use your CAS calculator to integrate the function. |  |  |\n| \u003Cbr>Rules for Anti-Differentiation\u003Cbr>\u003Cbr>• ∫ (􀜽􀝔 + 􀜾)􀯡 􀝀􀝔 = (􀯔􀯫􀯔)􀳙11 + c, where 􀝊 ≠ −1\u003Cbr>• ∫ 1􀯫 􀝀􀝔 = ∫ 􀝔 −1 􀝀􀝔 = log􀯘 (􀝔 ) + 􀜿\u003Cbr>• ∫ ~~ ~~(􀯔􀯫􀯕) 􀝀􀝔 = ∫ (􀜽􀝔 + 􀜾)−1 􀝀􀝔 = 1􀯔 log 􀯘 (􀜽􀝔 + 􀜾) + 􀜿\u003Cbr>• ∫ 􀝁 􀯫 􀝀􀝔 = 􀝁 􀯫 + 􀜿\u003Cbr>• ∫ 􀝁 􀯞􀯫 􀝀􀝔 = 1􀯞 􀝁 􀯞􀯫 + 􀜿\u003Cbr>• ∫ sin(􀜽􀝔) 􀝀􀝔 =  cos 􀜽􀝔 + 􀜿\u003Cbr>• ∫ sin(􀜽􀝔 + 􀜾) 􀝀􀝔 = − ~~ ~~1􀯔 cos(􀜽􀝔 + 􀜾) + 􀜿\u003Cbr>• ∫ cos(􀜽􀝔) 􀝀􀝔 = 1􀯔 sin 􀜽􀝔 + 􀜿\u003Cbr>• ∫ cos(􀜽􀝔 + 􀜾) 􀝀􀝔 = 1􀯔 sin 􀜽􀝔 + 􀜾  + 􀜿\u003Cbr>\u003Cbr>|  |  |  |\n|  | Find the antiderivative of  3 \u003Cbr>􀯫\u003Cbr>∫~~ ~~3􀝔 − ~~ ~~ 􀝀􀝔= ∫ 3􀝔1 − ~~ ~~~~ ~~ 􀝀􀝔\u003Cbr>\u003Cbr>\u003Cbr>1\u003Cbr>− ~~ ~~\u003Cbr>\u003Cbr>2􀯫−2\u003Cbr>\u003Cbr>3\u003Cbr>= ~~ ~~\u003Cbr>12\u003Cbr>1 1\u003Cbr>􀝔 2 −  2 􀝈􀝋􀝃􀯘 2􀝔 − 2 + 􀜿\u003Cbr>\u003Cbr>1 1\u003Cbr>= 6􀝔2 −  2 􀝈􀝋􀝃􀯘 2􀝔 − 2 + 􀜿= 6 − ~~ ~~12 􀝈􀝋􀝃􀯘  + 􀜿\u003Cbr>|  |  |\n\n\n| \u003Cbr>If 􀝂 is a continuous function on the interval [a, b] then\u003Cbr>􀯕\u003Cbr>∫ 􀝂 (􀝔) 􀝀􀝔 = [􀜨  ]􀯕􀯔􀯔\u003Cbr>= 􀜨 􀜾 − 􀜨 􀜽\u003Cbr>Where 􀜨 is the antiderivative of 􀝂 .\u003Cbr>|  | \u003Cbr>• ∫􀯔􀯔 􀝂  􀝀􀝔 = 0\u003Cbr>• ∫􀯕􀯔 􀝂 (􀝔) 􀝀􀝔 = − ∫􀯕􀯔 􀝂  􀝀􀝔\u003Cbr>• ∫􀯕􀯔 􀝇􀝂 􀝀􀝔 = 􀝇 ∫􀯕􀯔 􀝂 (􀝔 ) 􀝀􀝔\u003Cbr>• ∫􀯕􀯔 (􀝂 ± 􀝃  ) 􀝀􀝔 = ∫􀯕􀯔 􀝂  􀝀􀝔 + ∫􀯕􀯔 􀝃 (􀝔 ) 􀝀􀝔\u003Cbr>• ∫􀯕􀯔 􀝂 (􀝔) 􀝀􀝔 = ∫􀯖􀯔 􀝂 (􀝔) 􀝀􀝔 + ∫􀯖􀯕 􀝂 (􀝔) 􀝀􀝔, providing\u003Cbr>􀜽 \u003C 􀜿 \u003C 􀜾 |\n| --- | --- | --- |\n\n[www. Maths](www. MathsMethodsClub.com.au)[Methods](www. MathsMethodsClub.com.au)[Club.com.au](www. MathsMethodsClub.com.au) 24 [Facebook.com/Maths](Facebook.com/MathsMethodsClub)[Methods](Facebook.com/MathsMethodsClub)[Club](Facebook.com/MathsMethodsClub)","cbCaisKvhpkzblpw","https://ap.wps.com/l/cbCaisKvhpkzblpw","pdf",279457,"English","# Integration\n## Properties of Integrals\n## Forgetting to include the constant\n## Rules for Anti-Differentiation\n## Definite Integral via an Antiderivative","[{\"question\":\"Why must the constant of integration be included after integrating?\",\"answer\":\"The notes highlight that forgetting to add the constant leads to incomplete antiderivatives. Add “+ c” after integrating to capture all possible solutions.\"},{\"question\":\"When do the antidifferentiation rules apply?\",\"answer\":\"They apply only when the expression inside the brackets is linear. For other kinds, expand if possible before integrating or use a CAS calculator.\"},{\"question\":\"How is a definite integral computed using an antiderivative?\",\"answer\":\"If the function is continuous on [a, b], the notes state that the definite integral equals the antiderivative evaluated at b minus its value at a, i.e., F(b) − F(a).\"}]","Maths-Methods-Club-Integration-Cheat-Sheet-Sample - Anti-Differentiation & Definite Integrals Rules | PDF",1791929912]