[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572935-105":53,"doc-detail-572935-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","ch320methods20in20calculus-chapter-3-calculus-methods","Ch.3%20Methods%20in%20Calculus - Chapter 3 - Calculus Methods","","This chapter extends earlier calculus techniques by focusing on improper integrals, differentiation and integration of inverse trigonometric functions, and partial-fraction methods for integrating rational expressions with quadratic factors in the denominator. It explains when an improper integral is evaluable, distinguishes convergence from divergence, and outlines limiting procedures when limits are infinite or when the integrand is undefined within the interval. Worked examples develop these ideas and apply mean value results for functions on an interval.",{"@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/ch320methods20in20calculus-chapter-3-calculus-methods/572935/",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/ch320methods20in20calculus-chapter-3-calculus-methods/572935.png","ImageObject",442,249,{"name":88,"@type":89},"supergirl","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},"How do you determine whether an improper integral is convergent or divergent?","Question",{"text":108,"@type":109},"An improper integral is convergent if the limiting value exists after applying the required limiting process. If the limit does not exist, the integral is divergent and cannot be evaluated.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What steps are used when one integration limit is infinite?",{"text":113,"@type":109},"Replace infinity with another variable, evaluate the integral normally, then take the limit of the resulting expression as the variable approaches ±∞, depending on which side the range goes.",{"name":115,"@type":106,"acceptedAnswer":116},"How can you differentiate inverse trigonometric functions like arcsin, arccos, and arctan?",{"text":117,"@type":109},"Use the provided derivative results, which can be proved via implicit differentiation. For derivatives of compositions such as arcsin(2x), apply implicit differentiation or the chain rule rather than substituting directly into the base formulas.","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},572935,1792053579,{"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":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},962088121634,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Methods in Calculus Cheat Sheet  \nThe aim of this chapter is to extend upon the calculus you learnt in Pure Year 2. We will look at improper integrals, the differentiation and integration of inverse trigonometric functions and the use of partial fractions to integrate fractional expressions with quadratic factors in the denominator. Proficiency with the differentiation and integration techniques you covered in Pure Year 2 will be key in this chapter.  \nDefining improper integrals  \nYou need to know what is meant by an improper integral, and when they can be evaluated.  \n􀂃 The integral 􀶱􀯔􀯕􀝂 (􀝔) 􀝀􀝔 is improper if: 􀜽 ≤ 􀝔 ≤ 􀜾 could also be written as [􀜽, 􀜾]  \no one or both limits are infinite.   \no 􀝂 (􀝔) is undefined at some point in the interval 􀜽 ≤ 􀝔 ≤ 􀜾 .  \nIf an improper integral exists, we say it is convergent. Otherwise we say it is divergent.  \nThe region R is represented by the improper integral  \n􀶱2∞ 􀝔12  \n􀝀􀝔  \nThis is a improper integral because the upper limit of the integral is positive infinity.  \nThe region R is represented by the improper integral  \n􀶱02 􀝔  \n􀝀􀝔  \nThis is an improper integral because the range of integration [0,2] contains the point 􀝔 = 0, where the curve is undefined since  \n􀝔 = 0 is an asymptote.  \nIntegrals where one of the limits is infinite  \nWhen evaluating integrals with infinity in one of the limits, we use a process involving limits:  \n1. Replace the infinity with another variable (e.g. 􀝐)  \n2. Evaluate the integral as usual.  \n3. Consider the limit of your result as 􀝐 → ±∞ (this will depend on whether the range of integration included positive or negative infinity) .  \nThe limit of the result will give you the final answer. It is important to note that this limit will not always exist. In such cases, the integral is said to be divergent and cannot be evaluated. If the limit does exist however, the integral is convergent and we can evaluate it.  \nExample 1: Evaluate the integral 􀶱2∞ 􀝔12 􀝀􀝔 , or show that it is not convergent.  \n\n| Replacing the infinity with 􀝐 | 􀶱2􀯧 􀝔12 􀝀􀝔 |  |  |  |\n| --- | --- | --- | --- | --- |\n| Carrying out the integration: | 􀵤 − 1􀝔􀵨 |  | 􀯧\u003Cbr>2 | = 􀵤 − ~~ ~~1􀝐 + 12􀵨 |\n| Finding the limit of our result as 􀝐 → ∞ Note that as 􀝐 → ∞, − 1􀯧 → 0. | lim 􀯧→∞ | 􀵤 − ~~ ~~1􀝐 + 12􀵨 = 12 |  |  |\n\nIntegrals where both limits are infinite  \nIf both limits are infinite, then you need to split the integral up into the sum of two improper integrals as follows:  \n􀂃 􀶱− 􀝂 (􀝔) 􀝀􀝔 = 􀶱−􀝂 (􀝔) 􀝀􀝔 + 􀶱􀯞∞􀝂 (􀝔) 􀝀􀝔  \nYou can choose any value of 􀝇 when splitting up such an integral. Remember that if both integrals converge then the original will converge. If one diverges however, then the original will diverge.  \nIntegrals that are undefined at some point  \nWhen dealing with integrals where the integrand is undefined at some point in the given interval, we use a very similar limiting process. If the undefined point is one of your limits of integration:  \n1. Identify which point, 􀝇 , in the interval is undefined and replace this with another variable, 􀝐 .  \n2. Evaluate the integral as usual.  \n3. Consider the limit of your result as 􀝐 → 􀝇 . This will be your final answer.  \nIn the rare occasion that the undefined point is not one of your limits, you will need to split the integral up before using the above process:  \n􀂃 If 􀝂(􀝔) is undefined at 􀝔 = 􀝇 over the interval [􀜽, 􀜾], then 􀶱􀯔􀯕 􀝂 (􀝔) 􀝀􀝔 = 􀶱􀯔􀯞􀝂 (􀝔) 􀝀􀝔 + 􀶱􀯞􀯕􀝂 (􀝔) 􀝀􀝔 Example 2: Evaluate the integral 􀶱02 􀝔 􀝀􀝔, or show that it is not convergent.  \n\n| Notice that the function is undefined at 􀝔 = 0, so we replace 0 with 􀝐 and integrate. | 􀶱􀯧 2 􀝔 􀝀􀝔 = 􀵣2~~√~~􀝔􀵧 | 2 􀯧 | = 􀵣2√2 − 2√􀝐􀵧 |\n| --- | --- | --- | --- |\n| Finding the limit of our result as 􀝐 → 0. Note that as 􀝐 → 0, 2√􀝐 → 0. | \u003Cbr>lim􀵣2√2 − 2√􀝐􀵧 = 2√2 + 0 = 2√2 􀯧→0 |  |  |\n\nThe mean value of a function  \nYou already know that to calculate the mean of a set of values, you must sum up the values and divide by the number of values. We will now extend this idea to look at how we find the mean value of ","cbCaifMpcZrFh1Pa","https://ap.wps.com/l/cbCaifMpcZrFh1Pa","pdf",420288,"English","# Defining improper integrals\n## When an improper integral is improper\n## Infinite limits and convergence/divergence\n## Both limits infinite\n## Integrals undefined at a point\n# Mean value of a function\n## Mean value rules for transformed functions\n# Differentiating inverse trigonometric functions\n## Core derivative formulas and method notes","[{\"question\":\"How do you determine whether an improper integral is convergent or divergent?\",\"answer\":\"An improper integral is convergent if the limiting value exists after applying the required limiting process. If the limit does not exist, the integral is divergent and cannot be evaluated.\"},{\"question\":\"What steps are used when one integration limit is infinite?\",\"answer\":\"Replace infinity with another variable, evaluate the integral normally, then take the limit of the resulting expression as the variable approaches ±∞, depending on which side the range goes.\"},{\"question\":\"How can you differentiate inverse trigonometric functions like arcsin, arccos, and arctan?\",\"answer\":\"Use the provided derivative results, which can be proved via implicit differentiation. For derivatives of compositions such as arcsin(2x), apply implicit differentiation or the chain rule rather than substituting directly into the base formulas.\"}]","Ch.3%20Methods%20in%20Calculus - Chapter 3 - Calculus Methods | PDF",1791930163]