[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571592-105":53,"doc-detail-571592-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","integration-cheat-sheet-math-study-guide","Integration Cheat Sheet - Math Study Guide","","Integration is the inverse of differentiation and is used to find areas enclosed between curves and coordinate axes, extending well beyond area problems into areas such as Physics, Statistics, and Economics. The sheet consolidates key standard results for integrals, including trigonometric and rational forms, and highlights derivations from a differentiation formula booklet. It also explains integration strategies such as identities, “by parts,” partial fractions, and trapezium-rule numerical estimation, including concave/convex error direction and percentage error, plus a brief substitution method example for more complex integrals.",{"@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":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/integration-cheat-sheet-math-study-guide/571592/",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/integration-cheat-sheet-math-study-guide/571592.png","ImageObject",442,249,{"name":88,"@type":89},"Adam","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How is integration related to differentiation, and what does it help you find?","Question",{"text":108,"@type":109},"Integration is the inverse of differentiation. It is primarily used to find areas enclosed between curves and coordinate axes, and it also supports problems in other disciplines such as Physics, Statistics, and Economics.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"When you cannot integrate a trigonometric expression directly, what approach should you use?",{"text":113,"@type":109},"Manipulate the expression into a form you can integrate by using the trigonometric identities from earlier chapters, such as transforming sin powers using standard identities.",{"name":115,"@type":106,"acceptedAnswer":116},"How do you estimate an integral numerically using the trapezium rule, and how do you judge error direction?",{"text":117,"@type":109},"Divide the region into vertical strips and approximate each strip’s area using the trapezium formula, then sum them. Determine whether the curve is convex (overestimate) or concave (underestimate) to infer the direction of the estimate’s error, and compute % error using the given formula.","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},571592,1792250978,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"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},1374404737137,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Integration Cheat Sheet  \nIntegration is the inverse of differentiation. We can think of integration as a mathematical tool that allows us to find areas enclosed between curves and the coordinate axes. In fact, the uses of integration extend far beyond finding areas and can also be found in other fields of study, including Physics, Statistics and even Economics.  \nVery useful results  \nf(􀝔)  \n􀝔n  \neax+b  \n1  \na􀯫+b  \n(a􀝔+b)n  \nsin k􀝔  \ncos k􀝔  \n∫f(􀝔) d􀝔  \n􀯫􀳙+1  \n􀯡+1  \n1a eax+b 1a ln|a􀝔+b| 1 ሺax+bሻ􀝊+1 a n+1 − 1k cos k􀝔 1k sin k􀝔  \nThese can be derived from the differentiation section in the formula booklet  \nf(􀝔)  \ntan k􀝔 cot k􀝔  \ncosec k􀝔  \nsec k􀝔  \nsec2 k􀝔  \nsec k􀝔 tan k􀝔  \ncosec2 k􀝔  \ncosec k􀝔 cot k􀝔  \n∫f(􀝔) d􀝔  \n1􀯞 ln|sec k􀝔|  \n1􀯞 ln|sin k􀝔|− 1􀯞 ln|cosec k􀝔 + cot k􀝔| 1􀯞 ln|sec k􀝔 + tan k􀝔| 1􀯞 tan k􀝔  \n1􀯞 sec k􀝔− 1􀯞 cot k􀝔  \n1  \n− ~~ ~~  \n􀯞  \ncosec k􀝔  \n“by parts” formula:  \nTrapezium rule:  \nYou will also be given these in the formula booklet.  \nJust like in Pure Year 1, you will need to use integration to find areas. You should remember:  \n▪ ∫􀯔􀯕 􀝕 􀝀􀝔 represents the area bounded between the curve  \n􀝕 = 􀝂ሺ􀝔ሻ, the 􀝔-axis and the lines 􀝔 = 􀜽 and 􀝔 = 􀜾 .  \n▪ When a function is given parametrically, the area under the curve is given by ∫􀯔􀯕 ቂ􀝕 􀯗􀯫􀯗􀯧ቃ 􀝀􀝐 . Remember that 􀜽 and 􀜾 are limits given in terms of 􀝐, and the integration is done with respect to 􀝐 .  \n▪ You need to be able to recognise that  \nlim  \n􀰋􀯫→0  \n􀜾∑ 􀝂ሺ􀝔ሻ􀟜􀝔􀝔 =􀜽  \n= 􀯕􀝂ሺ􀝔ሻ 􀝀􀝔  \nYou might come across an expression involving trigonometric functions that you can’t integrate straight away using one of the results above. For example, ∫sin3 􀝔 d􀝔 can’t be found directly.  \nTo tackle problems of this sort, you need to manipulate the expression into a form you can integrate. This is why it is crucial you are familiar with all of the identities you encountered in chapters 6 and 7. Below is an example showcasing we do this in practice.  \nExample 1: Evaluate ∫ sin2 􀝔 d􀝔  \n∫ sin2 􀝔 􀝀􀝔 = ∫ ሺ1 − 􀜿􀝋􀝏 2􀝔ሻ􀝀􀝔 = ∫ 1 􀝀􀝔 − ∫ 􀜿􀝋􀝏 2􀝔 􀝀􀝔 .􀜤􀝑􀝐 􀜿􀝋􀝏2􀝔 = 2􀜿􀝋􀝏 2􀝔 − 1, 􀝏􀝋 􀜿􀝋􀝏 2􀝔 = 􀜿􀝋􀝏2􀝔2~~ ~~+~~ ~~1  \n∴ ∫ 􀜿􀝋􀝏 2􀝔 􀝀􀝔 = 12 ∫ 􀜿􀝋􀝏2􀝔 + 1 􀝀􀝔 = 12 [12 􀝏􀝅􀝊2􀝔 + 􀝔] = 14 􀝏􀝅􀝊2􀝔 + 12 􀝔􀝏􀝋 ∫ 􀝏􀝅􀝊2 􀝔 􀝀􀝔 = 􀝔 − 14 􀝏􀝅􀝊2􀝔 − 12 􀝔 + 􀜿  \nusing 􀝏􀝅􀝊2x = 1 − 􀜿􀝋􀝏 2x  \nusing the double angle formulae 􀜿􀝋􀝏2􀝔 = 2􀜿􀝋􀝏 2􀝔 − 1  \nSome complicated expressions can be integrated very easily if they are of one of the forms below:  \n[1] ∫ 􀝂′ ሺ􀝔ሻ [􀝂ሺ􀝔ሻ]􀯡 􀝀􀝔 = 􀯙ሺ􀯫􀯡1+1 + 􀜿 . e.g. ∫ 􀝏􀝅􀝊400􀝔 􀜿􀝋􀝏􀝔 􀝀􀝔 = 􀯦􀯜􀯡440011􀯫 + 􀜿∫ 􀝔2√􀝔 3 + 4 􀝀􀝔 = 13 ∙ ~~ ~~ሺ􀯫34ሻ32~~ ~~  \n[2] ∫ 􀯙􀯙′ሺ􀯫ሺ􀯫ሻሻ 􀝀􀝔 = ln|􀝂ሺ􀝔ሻ| + 􀜿 e.g. ∫ 􀯫2242 􀝀􀝔 = ln|􀝔2 + 42| + 􀜿  \n∫ 􀯦2􀯜􀯖􀯡􀯫􀯢􀯦􀯖2􀯢􀯫1 􀝀􀝔 = ∫ 21~~2~~􀯖􀯦􀯢􀯜􀯦􀯡22􀯫1 􀝀􀝔 = 12 ቂ − 14 􀝈􀝊|2􀜿􀝋􀝏2􀝔 + 1|ቃ + 􀜿  \nRemember that you must adjust for any variation in constants. In example 2 of rule 1, we had to multiply our answer by 13 since the differential of ሺ􀝔3 + 4ሻ5 is 3􀝔2 , not 􀝔2 .  \nWhile it is true you do not necessarily need to know the above rules, it is still very worthwhile for you to take the time to learn to apply them because they can greatly simplify otherwise difficult integrals.  \nYou could also be asked to integrate an expression involving a fraction with more than one linear factor in the  \n 11􀯫2 +14􀯫+5   \ndenominator. Take for example, the expression ሺ􀯫+1ሻ2ሺ2􀯫+1ሻ . This cannot be integrated directly so we need  \nto simplify it, which is where partial fractions come in handy. Using partial fractions, we can rewrite this as  \n 4  −  2  +  3   \n􀯫+1 ሺ􀯫+1ሻ2 ሺ2􀯫+1ሻ .  \nNow, each of the above terms can be integrated directly using one of the standard results on the left. Carrying out the integration:  \n∫ ሺ1􀯫21ሻ+21ሺ42􀯫􀯫51ሻ 􀝀􀝔 = ∫ 􀯫~~ ~~1 − ሺ􀯫1ሻ2 + ሺ2􀯫3+1ሻ 􀝀􀝔= 4 ln|􀝔 + 1| + 􀯫1 + 32 ln|2􀝔 + 1| +c  \nSuch questions are often worth upwards of 6 marks and are heavily reliant on your ability to use partial fractions well. The integration is mostly straight forward and does not require much extra work outside of applying the standard results, but it is the partial fractions procedure that can be a little tedious.  \nIn some cases, you might not be able to ","cbCaiiMjXF1MIRn5","https://ap.wps.com/l/cbCaiiMjXF1MIRn5","pdf",944203,"English","# Integration Cheat Sheet\n## Core idea and applications\n## Useful standard integration results\n## Deriving results from differentiation booklet\n## Area under curves and definite integral interpretation\n## Trigonometric integration strategy (identities)\n## Example: evaluating an integral with power-reduction\n## Integration patterns and constant adjustment\n## Partial fractions for rational functions\n## Trapezium rule for numerical integration\n## Overestimate vs underestimate using concavity\n## Percentage error in numerical estimates\n## Integration via substitution (intro)","[{\"question\":\"How is integration related to differentiation, and what does it help you find?\",\"answer\":\"Integration is the inverse of differentiation. It is primarily used to find areas enclosed between curves and coordinate axes, and it also supports problems in other disciplines such as Physics, Statistics, and Economics.\"},{\"question\":\"When you cannot integrate a trigonometric expression directly, what approach should you use?\",\"answer\":\"Manipulate the expression into a form you can integrate by using the trigonometric identities from earlier chapters, such as transforming sin powers using standard identities.\"},{\"question\":\"How do you estimate an integral numerically using the trapezium rule, and how do you judge error direction?\",\"answer\":\"Divide the region into vertical strips and approximate each strip’s area using the trapezium formula, then sum them. Determine whether the curve is convex (overestimate) or concave (underestimate) to infer the direction of the estimate’s error, and compute % error using the given formula.\"}]","Integration Cheat Sheet - Math Study Guide | PDF",1791922529]