[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-137733-105":59,"doc-detail-137733-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","chapter-13-additional-integration-topics","Chapter 13-Additional Integration Topics","","This document, likely a chapter from a mathematics textbook, delves into advanced integration techniques. It covers calculating the area between curves defined by functions of x, including cases where one function is consistently above the other. The text introduces and demonstrates the method of integration by parts, explaining its application when direct integration is not feasible and highlighting the strategic selection of f(x) and dv. Several examples illustrate the repeated use of integration by parts to solve complex integrals, such as those involving polynomial and exponential functions. The document also touches upon integration of logarithmic functions, providing worked-out examples. Furthermore, it explores substitution methods for simplifying integrals, such as integrating expressions involving (1+e^x) and logarithmic functions. The chapter systematically builds upon foundational integration concepts, equipping students with a comprehensive toolkit for tackling a wider range of integration problems.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/chapter-13-additional-integration-topics/137733/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/chapter-13-additional-integration-topics/137733.png","ImageObject",300,407,{"name":92,"@type":93},"Olivia Brown","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-18","2026-08-22",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the primary method discussed for solving complex integrals in this chapter?","Question",{"text":113,"@type":114},"The chapter primarily discusses and demonstrates the method of integration by parts, which is used when direct integration is not feasible.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is the area between two curves calculated when their functions intersect?",{"text":118,"@type":114},"For areas bounded by y=f(x) and y=g(x) where f(x) >= g(x) within the interval [a, b], the area is calculated by integrating the difference between the two functions, ∫[f(x) - g(x)] dx, from a to b.",{"name":120,"@type":111,"acceptedAnswer":121},"Can you provide an example of a repeated use of integration by parts?",{"text":122,"@type":114},"Yes, the document shows an example of integrating x^2 * e^(-x) using a repeated application of integration by parts, leading to the result -x^2*e^(-x) - 2xe^(-x) - 2e^(-x) + C.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},137733,1787439706,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":34,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":144},16904993612988,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","ma00  \n13-1-1  \nae /nao  \n13  \nTopic  \nte  \nCrea  \n13-1  \nbotween  \nCeVeS  \nby  \nCLrea  \nbo unded  \ny=fcx) and  \n=x  \nfoa七≤b  \n(X)≥x  \nwhene  \ny个f(xy=.9cx)wx  \narew A三。 f(x) d x bete (x  \nand  \nx)  \ndx  \n。f(x)-g(x),  \n二  \nf(ck)-xtn lesof hhighwidh  \nSwm  \nf  \n△X  \ny  \n→ -·  \nas  \nrea boundedX=-2,andx=1  \nEx  \n>0  \n13-1-2  \n= f(x)-(x]d=  \na ndedf(x)=9(x)byf(x=-x^2  nx=2-25-x²=2-2xx²-2x-3=0,x-3(x+1D=cntsectuemX=-1 X=3f(-x+=1)(x-3)in(一1,3),fcx)>g(x)=Lf(x)-9(x) dx∫(-x²+2x+3)dx3 L,+x²1233x-1  \n13-3-13- lntegraton pazts于d d    xx)+fx)f(x) fxcx)]-fx(xfangix de(fangrx ]dfing00dle f(x)g(x)一f(x,g(x d we   useF(x)drx=Fx十  \nfx)dv=gx)  \nfcx)  \ngx)  \ndvu - du  \nffx xdfcx)d   \nwhen  \nhs fo2mu lausedis impossible to  integrate  Canlbe integrated  \nis  \nwhile  \n2l=fcX)  \nchoser—with  \ncw2  \n=X  \n ntgra  \nis teintgrated  \nfcx) gCx) dxgdxCan  \n①  \nbe  \n②  \n13-3-2  \nEX  \nxe* dx  \nf=七f=gcx)=ex=f(x gxd=fx g(x-iangadle xex-dx=eexedx=x=xe×-fexdG)^= xe×fcx)gx↓e×dx=xeˣ-ex+C↓)fx)  \nnXd  \nE×  \nf(x)=lnx  \ngX)=x  \nnxdx=f1xgx)dxx()一fdx=x²lnxx²+  \nxlnx dx=fLnx d(二=nx-d(lne  \n13-3-3  \nRpeate   se  of intyratin  p ass  \n^2e=x²(-e⁻×)’dx  \n三-x²e⁻×-c-e×)(x²)'dx一+xe-dx  \nxe×dxJ=x (e×)dx  \n=-xe×+e×dx=-xe×(-e×)(x)³dz=-xe⁻×-e*J+C=e“du=“-  \nwher  \nx²exdx  \n=-x²e×+2 -xe×-e×+c)  \n三一x^-2x一  \nFor foniteintgrsnxlnx dx二 nx)七x(n=xlnXxdx = xnx- ox一=xlnx-x+Cnxd化=(ln-(e lne-e)-(1 ln1-1)二  \nMe2eamplq①ig13-3-4“ d= ce)e-e (x 寸一2x dx-e f三x²e×=x²e×-2xe上fxd(ex)=x²ex-xe+2fe×dx+2e×+C  \n③  \nfeLxd  \nJ  \nn+七dx+xxd u/nC1+x^2)  d )+x2Cnu  du, de  \ndu= u ln u-u d cnu)=ulna-idulnu-du= uln u-u+C  \n+x2n一x十  \n13-3-5  \ne× (1+e×) dx二Cn(1+ e×)dlex+1)Jln u du2>三uln u-U+C=(1+e×)CnCl+e×)-(1+e×)+  \n⑤nx)=x(Cnx)²-Jx[(Cnx)²]dafcx)d=xfcxxfix)dth (x)=x(nx)²]'=2Lnx&Gnx=交Gnxdx=ndx= dx=xd=2 xnx-2x+Cx(lnx)²-2xmx+2X+","cbCaiq2RYYhcvZjp","https://ap.wps.com/l/cbCaiq2RYYhcvZjp","pdf",604027,"English","# Chapter 13-Additional Integration Topics\n## 13-1 Area Between Curves\n### 13-1-1 Area bounded by y=f(x) and y=g(x)\n### 13-1-2 Area bounded by f(x)=g(x)\n## 13-3 Integration by Parts\n### 13-3-1 Integration by Parts Formula\n### 13-3-2 Example 1\n### 13-3-3 Example 2\n### 13-3-4 Repeated Use of Integration by Parts\n### 13-3-5 Example 1","[{\"question\":\"What is the primary method discussed for solving complex integrals in this chapter?\",\"answer\":\"The chapter primarily discusses and demonstrates the method of integration by parts, which is used when direct integration is not feasible.\"},{\"question\":\"How is the area between two curves calculated when their functions intersect?\",\"answer\":\"For areas bounded by y=f(x) and y=g(x) where f(x) \\u003e= g(x) within the interval [a, b], the area is calculated by integrating the difference between the two functions, ∫[f(x) - g(x)] dx, from a to b.\"},{\"question\":\"Can you provide an example of a repeated use of integration by parts?\",\"answer\":\"Yes, the document shows an example of integrating x^2 * e^(-x) using a repeated application of integration by parts, leading to the result -x^2*e^(-x) - 2xe^(-x) - 2e^(-x) + C.\"}]","Chapter 13-Additional Integration Topics | PDF",18]