[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572895-105":53,"doc-detail-572895-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","techniques-of-integration-8-1-substitution","Techniques of Integration - 8. 1 Substitution","","Techniques for finding antiderivatives focus on recognizing when an integrand is already a straightforward derivative, allowing immediate antiderivative rules and quick constant adjustments. The section extends to chain-rule patterns, showing how to verify results by differentiation. It develops substitution as a practical method: choosing an inner function candidate u, rewriting the integrand completely in terms of u, integrating, and then converting back to x. Worked examples illustrate standard substitutions and evaluation steps.",{"@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/techniques-of-integration-8-1-substitution/572895/",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/techniques-of-integration-8-1-substitution/572895.png","ImageObject",442,249,{"name":88,"@type":89},"Alex Sinclair","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-17","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":76},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How can an integrand be identified as a simple derivative?","Question",{"text":108,"@type":109},"If the integrand matches a derivative form directly, its antiderivative can be written immediately from known derivative rules. Constant factors are adjusted afterward since differentiation often ignores constants in certain contexts.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How does the chain rule guide integration when the integrand is not an obvious derivative?",{"text":113,"@type":109},"Look for a “inside function” whose derivative appears as a factor in the integrand. If differentiating a proposed antiderivative produces the given integrand via the chain rule, the result is confirmed.",{"name":115,"@type":106,"acceptedAnswer":116},"What is the substitution method and how is it applied?",{"text":117,"@type":109},"Choose a likely inner function u and rewrite the integrand entirely in terms of u, eliminating x from the expression. Compute the integral in u, then substitute back using the relationship between u and x.","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},572895,1792123981,{"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":76,"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":141},1099523882182,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","8  \nTechniques of Integration  \nOver the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. Sometimes this is a simple problem, since it will be apparent that the function you wish to integrate is a derivative in some straightforward way. For example, faced with  \nZ x 10 dx  \nwe realize immediately that the derivative of x 11 will supply an x10 : (x11 )′ = 11x10 . We don’t want the “11”, but constants are easy to alter, because di􀀋erentiation “ignores” them in certain circumstances, so  \nddx 111 x 11 = 111 11x10 = x 10 .  \nFrom our knowledge of derivatives, we can immediately write down a number of antiderivatives. Here is a list of those most often used:  \nZxn dx = xn11 + C, if n  −1  \nZ x −1 dx = ln|x| + CZ ex dx = ex + C  \nZ sin x dx = − cos x + C  \n164 Chapter 8 Techniques of Integration  \nZ cos x dx = sin x + CZ sec2 x dx = tan x + CZ sec x tan x dx = sec x + CZ 1~~ ~~x2 dx = arctan x + CZ √~~ ~~1~~ ~~~~ ~~x2 dx = arcsin x + C  \nNeedless to say, most problems we encounter will not be so simple. Here’s a slightly more complicated example: 􀀌nd  \nZ 2x cos(x2 ) dx.  \nThis is not a “simple” derivative, but a little thought reveals that it must have come from an application of the chain rule. Multiplied on the “outside” is 2x, which is the derivative of the “inside” function x2 . Checking:  \nddx sin(x2 ) = cos(x2 ) ddxx2 = 2x cos(x2 ) ,  \nso  \nZ 2x cos(x2 ) dx = sin(x2 ) + C.  \nEven when the chain rule has “produced” a certain derivative, it is not always easy to see. Consider this problem:  \nZ x3 p 1 − x2 dx.  \nThere are two factors in this expression, x3 and p 1 − x2  , but it is not apparent that the chain rule is involved. Some clever rearrangement reveals that it is:  \nZ x3 p 1 − x2 dx = Z (−2x) 􀀒 − ~~ ~~12􀀓 (1 − (1 − x2 )) p1 − x2 dx.  \nThis looks messy, but we do now have something that looks like the result of the chain rule: the function 1 − x2 has been substituted into −(1/2)(1 − x)√x, and the derivative  \n8.1 Substitution  \n8. 1 Substitution 165  \nof 1 − x2 , −2x, multiplied on the outside. If we can 􀀌nd a function F(x) whose derivative  \nis −(1/2)(1 − x)√x we’ll be done, since then  \nddxF(1 − x2 ) = −2xF′(1 − x2 ) = (−2x) 􀀒 − ~~ ~~12􀀓 (1 − (1 − x2 )) p1 − x2  \n= x3 p1 − x2  \nBut this isn’t hard:  \nZ − ~~ ~~12 (1 − x)√x dx = Z − ~~ ~~12 (x1/2 − x3/2) dx (8 .1. 1)  \n= − ~~ ~~12 􀀒 23x3/2 − 25x5/2􀀓 + C  \n= 􀀒 15x − 13􀀓 x3/2 + C.  \nSo 􀀌nally we have  \nZ x3 p1 − x2 dx = 􀀒 15 (1 − x2 ) − 13􀀓 (1 − x2 )3/2 + C.  \nSo we succeeded, but it required a clever 􀀌rst step, rewriting the original function so that it looked like the result of using the chain rule. Fortunately, there is a technique that makes such problems simpler, without requiring cleverness to rewrite a function in just the right way. It sometimes does not work, or may require more than one attempt, but the idea is simple: guess at the most likely candidate for the “inside function”, then do some algebra to see what this requires the rest of the function to look like.  \nOne frequently good guess is any complicated expression inside a square root, so westart by trying u = 1 − x2 , using a new variable, u, for convenience in the manipulations that follow. Now we know that the chain rule will multiply by the derivative of this inner function:  \ndu  \n= −2x,  \ndx  \nso we need to rewrite the original function to include this:  \nZx3 p 1 − x2 = Zx3 √u ~~ ~~−2x2x dx = Z 22√u dudx dx.  \nRecall that one bene􀀌t of the Leibniz notation is that it often turns out that what looks like ordinary arithmetic gives the correct answer, even if something more complicated is  \n166 Chapter 8 Techniques of Integration  \ngoing on. For example, in Leibniz notation the chain rule is  \ndy   dy dt = dx dt dx .  \nThe same is true of our current expression:  \nZ 22√u dudx dx = Z 22√u du.  \nNow we’re almost there: since u = 1 − x2 , x2 = 1 − u and the integral is  \nZ − ~~ ~~12 (1 − u)√u du.  \nIt’s no coincidence that this is exactly ","cbCaicDXvQcspnQn","https://ap.wps.com/l/cbCaicDXvQcspnQn","pdf",200393,26,"English","# Techniques of Integration\n## Substitution\n## Key derivative-based antiderivatives\n## Chain-rule identification and verification\n## Substitution procedure with u-substitution\n## Example 8.1.1\n## Example 8.1.2\n## Example 8.1.3","[{\"question\":\"How can an integrand be identified as a simple derivative?\",\"answer\":\"If the integrand matches a derivative form directly, its antiderivative can be written immediately from known derivative rules. Constant factors are adjusted afterward since differentiation often ignores constants in certain contexts.\"},{\"question\":\"How does the chain rule guide integration when the integrand is not an obvious derivative?\",\"answer\":\"Look for a “inside function” whose derivative appears as a factor in the integrand. If differentiating a proposed antiderivative produces the given integrand via the chain rule, the result is confirmed.\"},{\"question\":\"What is the substitution method and how is it applied?\",\"answer\":\"Choose a likely inner function u and rewrite the integrand entirely in terms of u, eliminating x from the expression. Compute the integral in u, then substitute back using the relationship between u and x.\"}]","Techniques of Integration - 8. 1 Substitution | PDF",1791929921,9]