[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-572894-105":3,"detail-sidebar-cat-1-en-105":80,"doc-detail-572894-en":126},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","integration-chapter-13-integration-cheat-sheet","Integration - Chapter 13 - Integration Cheat Sheet","","Integration is presented as the reverse of differentiation, emphasizing how to compute areas bounded by curves and coordinate axes. The guide explains key notation (the integral symbol, integrand, and limits), distinguishes indefinite from definite integrals, and highlights the role of the constant of integration. It provides worked examples for evaluating common integral forms, handling polynomials, determining the constant from a point on a curve, and finding bounded areas, including cases where curves cross the x-axis.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/paper-templates/","Paper Templates",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/integration-chapter-13-integration-cheat-sheet/572894/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/integration-chapter-13-integration-cheat-sheet/572894.png","ImageObject",442,249,{"name":42,"@type":43},"Aria Callaghan","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":22},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What does the integral symbol represent, and what is the integrand?","Question",{"text":62,"@type":63},"The integral symbol represents integration. The expression between the integral sign and the differential is known as the integrand.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How do indefinite and definite integrals differ in their results?",{"text":67,"@type":63},"An indefinite integral includes an added constant of integration and yields a function, while a definite integral is bounded by two limits and yields a numerical value.",{"name":69,"@type":60,"acceptedAnswer":70},"Why must areas be split when a curve crosses the x-axis?",{"text":71,"@type":63},"When the curve is below the x-axis, the integral gives a negative area. If you combine positive and negative regions without splitting, they can cancel out and produce an incorrect total area.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},572894,1792057069,{"code":4,"msg":81,"data":82},"success",[83,88,93,98,103,108,113,118,122],{"id":84,"doc_module":22,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},11,"Presentations",90,"presentations",{"id":89,"doc_module":22,"doc_module_name":25,"category_name":90,"show_sort_weight":91,"slug":92},12,"Resumes",80,"resumes",{"id":94,"doc_module":22,"doc_module_name":25,"category_name":95,"show_sort_weight":96,"slug":97},14,"Invoices",70,"invoices",{"id":99,"doc_module":22,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},15,"Posters",60,"posters",{"id":104,"doc_module":22,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},16,"Social Media",50,"social-media",{"id":109,"doc_module":22,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},17,"Forms",40,"forms",{"id":114,"doc_module":22,"doc_module_name":25,"category_name":115,"show_sort_weight":116,"slug":117},18,"Letters",30,"letters",{"id":119,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":120,"slug":121},21,5,"papers-templates",{"id":123,"doc_module":22,"doc_module_name":25,"category_name":124,"show_sort_weight":4,"slug":125},158,"General","general-158",{"code":4,"msg":81,"data":127},{"doc_id":78,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":119,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":22,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":22,"language":135,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":12,"update_tm":139,"read_time":4},962084926284,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Integration Cheat Sheet  \nIntegration is the reverse of differentiation. We can use integration to find areas bounded between a curve and the coordinate axes.  \nNotation  \nThe ∫ symbol is used to represent integration. Since integration is the reverse of differentiation, we have that  \nThe 􀀡􀀢 means that we are integrating with respect to 􀀢  \n§ If !!\"\\# = 􀀢 (􀀤) then 􀀦 = ∫ 􀀢(􀀤) 􀀩􀀤  \nThe integral sign  \nThe expression we want to integrate goes  \nbetween the integral sign and the 􀀡􀀢 . This is  \nknown as the integrand.  \nIndefinite integrals  \nIn this chapter, you will only learn how to integrate functions of the form 􀀢!, where 􀀣 is a constant and 􀀣 ≠ −1 . To integrate functions of this form, you can use the following result:  \n§ ∫ 􀀢! 􀀡􀀢 = \"!!\"\\#\\#$ + 􀀪  \nThe \" + 􀀪\" is known as the constant of integration. To see why we need to add this constant to  \nour result, consider the following three functions:  \n- 􀀬 = 􀀢 % + 2  \n- 􀀬 = 􀀢 %  \n- 􀀬 = 􀀢 % − 9  \nIf we differentiate the above functions, the result is the same: &&'\" = 2􀀢 because the constant term disappears upon differentiating. But since integration is the reverse of differentiation, we should be able to integrate 2􀀢 and get back to whichever of those functions we started with. To allow for this, we must add the unknown constant of integration, 􀀪, to our end result.  \nThe above process is known as indefinite integration.  \nDefinite integrals  \nA definite integral is one where the integral is bounded between two limits. The main difference between a definite integral and an indefinite integral is that a definite integral will yield a numerical value, while an indefinite integral will yield a function. To calculate a definite integral:  \n§ ∫*) 􀀯 ((􀀢) 􀀲􀀢 = [􀀯(􀀢)]*) = 􀀯 (􀀵) − 􀀯(􀀶)  \nExample 1: Evaluate ∫%+ 􀀢 % 􀀲􀀢 .  \n7%+􀀢 % 􀀲􀀢  \n= 8􀀤33932 =  \nThe limits of the integral are 2 and 3, written on the top and bottom of the integral sign.  \nWe write the result in square brackets, with the limits outside  \n:333; − :233; = 9 − 83 = ~~ ~~193 We substitute our limits into the result, and simplify,  \nIntegrating polynomials  \nYou can use the following two properties of integrals to integrate expressions where there are multiple terms of the form 􀀰􀀤&  \n§ ∫ 􀀱􀀤& 􀀩􀀤 = 􀀱∫ 􀀤 & 􀀩􀀤 = 􀀱 2􀀬􀀭113 + 􀀪 for any constant 􀀼 .  \n§ ∫ 􀀤 & + 􀀤' 􀀩􀀤 = ∫ 􀀤 & 􀀩􀀤 + ∫ 􀀤 ' 􀀩􀀤  \nHere is an example showing how we use these properties in practice: Example 2: Evaluate ∫ 2􀀢+ − 2\") + 4􀀢 􀀲􀀢  \nWe can split up the integral as follows:  \n⇒ 2∫ 􀀢 + 􀀲􀀢 + (−4)∫ 􀀢 1% 􀀲􀀢 + 4∫ 􀀢 􀀲􀀢  \n⇒ 2 8􀀢429 + 􀀪 − 4 81$19 + 􀀲 + 4 8􀀢2%9 + 􀁉  \n⇒ 􀀢22 + 4􀀢 + 2􀀢% + 􀀼  \nEach of the integrals yields an integration constant (􀀪, 􀀲 and 􀁉), so we can denote their total as one constant, 􀀼 .  \nFinding the constant of integration  \nYou can find the constant of integration if you are given a point the curve passes through or the value of the function at a given point.  \nExample 3 : The curve 􀁊 with equation 􀁋 = 􀀿 (􀀢) passes through the point (4,5). Given that 􀀿& (􀀢) = '!~~√~~(~~'~~​%' , find the equation of 􀁊 .  \n􀀿 & (􀀢) = 􀀢􀀢$\"~~\\#~~ − 4􀀢$ . \\# = 􀀢 ~~ \"~~​! − 4􀀢  \n\\# !  \n􀀢 \" 4􀀢\" 2  \\# 8  !  \n∴ 􀀿(􀀢) =  52 − 32  + 􀁌 ⇒ 􀁋 = 5 􀀢 \" − 3 􀀢 \" + 􀁌  \nBut we are told that at 􀀢 = 4, 􀁋 = 5. Substituting these values:  \n2  \\# 8  !  \n5 = 5 (4)\" − 3 (4)\" + 􀁌  \n64 64 203 2  \\# 8  ! 203  \n5 =  5  −  3  + 􀁌 ⟹ 􀁌 =  15  ∴ 􀁋 = 5 􀀢 \" − 3 􀀢 \" +  15   \nFinding Areas  \nYou need to be able to use definite integration to find the area bounded between a curve and the 􀀢 -axis.  \n§ The area between a curve 􀀬 = 􀀯 (􀀢), the lines 􀀢 = 􀀶, 􀀢 = 􀀵 and the 􀀢-axis is given by  \n􀁆􀁊􀁉􀀶 = 7)*􀀯 (􀀢) 􀀲􀀢  \nAreas under the 􀀵-axis  \nWhen integrating over an interval where the curve is below the 􀀢-axis, the resultant area will be negative. As a result, you need to take extra care when finding areas under curves which are not strictly positive.  \n§ When integrating over an interval where the curve is both above and below the 􀀢-axis, you should split the integral up into separate regions where the function is strictly posit","cbCaibVut9zd6Aqi","https://ap.wps.com/l/cbCaibVut9zd6Aqi","pdf",264204,"English","# Integration Cheat Sheet\n## Notation\n## Indefinite integrals\n## Definite integrals\n## Integrating polynomials\n## Finding the constant of integration\n## Finding areas\n## Areas under the x-axis\n## More complicated areas","[{\"question\":\"What does the integral symbol represent, and what is the integrand?\",\"answer\":\"The integral symbol represents integration. The expression between the integral sign and the differential is known as the integrand.\"},{\"question\":\"How do indefinite and definite integrals differ in their results?\",\"answer\":\"An indefinite integral includes an added constant of integration and yields a function, while a definite integral is bounded by two limits and yields a numerical value.\"},{\"question\":\"Why must areas be split when a curve crosses the x-axis?\",\"answer\":\"When the curve is below the x-axis, the integral gives a negative area. If you combine positive and negative regions without splitting, they can cancel out and produce an incorrect total area.\"}]","Integration - Chapter 13 - Integration Cheat Sheet | PDF",1791929918]