[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-571593-105":3,"detail-sidebar-cat-1-en-105":81,"doc-detail-571593-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":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","integration-by-parts-cheat-sheet-liate","Integration By Parts Cheat Sheet - LIATE","","Integration by parts cheat sheet focused on when the integrand is a product of two unrelated functions and when reverse chain rule or substitution should be tried first. Covers the two harder families (natural logarithm and inverse trigonometry) by rewriting them into product form. Provides a step-by-step method, including choosing u and dv via LIATE, memorizing common templates, and an experienced “play it out” strategy to ensure the remaining integral becomes easier.",{"@graph":14,"@context":73},[15,34,56],{"@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/presentations/","Presentations",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/integration-by-parts-cheat-sheet-liate/571593/",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-by-parts-cheat-sheet-liate/571593.png","ImageObject",442,249,{"name":42,"@type":43},"Stanley","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-21","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",5,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How do you know when integration by parts is the right method?","Question",{"text":63,"@type":64},"Use parts when the integral looks like a product of two unrelated functions and substitution or reverse chain rule does not make the integrand easier. If substitution simplifies the integral after rewriting, it should be tried first.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What are the two harder integration types mentioned, and how should they be handled?",{"text":68,"@type":64},"Natural logarithm and inverse trigonometry cases are described as not being direct products and not obvious for parts. They need to be remembered as special cases and forced into a product form to apply integration by parts.",{"name":70,"@type":61,"acceptedAnswer":71},"What is LIATE and how is it used to choose u and dv?",{"text":72,"@type":64},"LIATE is an acronym used for beginners to decide which function to label as u and which as dv based on the function types’ order. The function that appears earlier on the LIATE list is chosen as u, and the other becomes the part to integrate accordingly.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},571593,1792359961,{"code":4,"msg":82,"data":83},"success",[84,88,93,98,103,108,113,118,122],{"id":85,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":86,"slug":87},11,90,"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":120,"show_sort_weight":55,"slug":121},21,"Paper Templates","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":82,"data":127},{"doc_id":79,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":85,"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":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":33,"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":26},2336477405376,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","[www.mymathscloud.com](www.mymathscloud.com) © MyMathsCloud  \nWhen the integral is a product of two completely unrelated functions (and we can’t use reverse chain rule or substitution) . Examples of this include:  \n• ∫ 􀀢􀀣􀀤􀀥􀀦 × 􀀨􀀦􀀩􀀪 e.g. ∫ 􀀫 􀀭􀀮􀀯 􀀫 􀀰􀀫 or ∫ 􀀫 !􀀭􀀮􀀯 􀀫 􀀰􀀫  \n• ∫ 􀀢􀀣􀀤􀀥􀀦 × 􀀱􀀲􀀳􀀣􀀴􀀥􀀴􀀵􀀩􀀶􀀷 e.g. ∫ 􀀫 􀀸 \" 􀀰􀀫 or ∫ 􀀫 ! 􀀸 \" 􀀰􀀫  \n• ∫􀀱􀀶􀀹􀀺 􀀳􀀣􀀤􀀥􀀦 × 􀀻􀀶􀀦􀀼􀀥􀀦 􀀳􀀣􀀤􀀥􀀦 e.g. ∫ 5􀀫√2 − 􀀫 􀀰􀀫  \n• ∫ 􀀢􀀣􀀤􀀥􀀦 × 􀁁􀀴 e.g. ∫ 􀀫 􀁂􀀯 􀀫 􀀰􀀫 or ∫ 􀀫 ! 􀁂􀀯 􀀫 􀀰􀀫 or ∫ \\#\" 􀁂􀀯 􀀫􀀰􀀫  \n• ∫ 􀀱􀀲􀀳􀀣􀀴􀀥􀀴􀀵􀀩􀀶􀀷 × 􀀨􀀦􀀩􀀪 e.g. ∫􀀸 \" sin 􀀫 􀀰􀀫 or∫ 􀀸 \" cos 􀀫 􀀰􀀫  \nThere are two harder types (natural logarithm and inverse trig) that aren’t a product and it is not obvious to use parts. These types need to be remembered as special cases and turned/forced into a product in order to use parts:  \n• ∫ln 􀀫 = ∫ 􀁊 × 􀀷􀀴 􀁋  \n• ∫ sin$\\# 􀀫 = ∫ 􀁊 × 􀀹􀀩􀀴$􀀥 􀁋  \n• ∫ cos $\\# 􀀫 = ∫ 􀁊 × 􀁌􀀣􀀹$􀀥􀁋  \n• ∫ tan$\\# 􀀫 = ∫ 􀁊 × 􀀵􀀶􀀴$􀀥 􀁋  \nA good rule of thumb to follow if you want to know whether to use parts would be to try a substitution first (so make sure you’re good at substitution before learning parts), and then if the resulting doesn’t become easier to integrate once simplified, try integration by parts. A substitution is mostly used (but certainly not always) whenever we have a function 􀁏 containing a function (we'll call this 􀁐) multiplied by the derivative of 􀁐 . That is, if you have the form ∫ 􀁐' (􀀫)􀁏 (􀁐(􀀫))􀀰􀀫 . For example,  \nS 􀀫􀀸 \"! 􀀰􀀫  \n􀁏 (􀁐 (􀀫)) = 􀀸 \"! 􀀭􀁔 􀁏 = 􀀸 \", 􀁐 = 􀀫 !  \nThe derivative of 􀀫 ! is 2􀀫 and we have the correct 􀀫 term. It doesn’t matter if the constant 2 in front is different. In this case we would use substitution (or reverse chain rule if you’re good at integration) instead of parts. It is worth mentioning that parts would still work, it would just take longer! So, lots of integrals that can be done by substitution can be done with parts instead, but the converse is not true.  \nThe best strategy is to try to \"play it out\" in your mind i.e. whether substitution or parts will be better and try to see which one will work better. The only way to get better at these sorts of integrals is to practice large sets of each type. Then, you start to think \"Oh—this looks like a substitution!\" or, \"maybe by-parts is better for  \nthis. \" Practise, practise, practiseis really the best way to get better at recognizing whether to use substitution or parts and is key to becoming good at parts.  \nWe say our original integral has the form/looks like ∫ 􀀢 !!\"\\# 􀀣􀀤 􀀦􀀧 ∫ !!\"\\# 􀀢 􀀣􀀤  \nStep 1:  \nCall one of the functions 􀁖 and the other function ((\") . It matters which we call 􀁖 and which we call ((\") !  \nSee the table on the bottom of the next page for how to choose which function is 􀁖 and which is ((\")  \nStep 2:  \nFind ((*\" by differentiating 􀁖 and 􀁗 by integrating ((\")  \nNotice how for both functions we want to end up with the original function and its corresponding derivative which leads us to step 3.  \n[www.mymathscloud.com](www.mymathscloud.com) © MyMathsCloud  \nStep 3:  \nPlug into the formula 􀁖􀁗 − ∫ 􀁗 ((*\" 􀀰􀀫  \nIt can help to remember the formula as ① − ②  \nStep 4:  \nSimplify the integral part ∫ 􀁗 ((*\" 􀀰􀀫 as much as possible before integrating. You take out constants (that are multiplied or divided) to make the integration easier.  \nStep 1 continued:  \nHow to choose which function is 􀁚 and which function is 􀀫􀀫􀀬􀀭 ?  \nWe need to call 􀁖 the function that we can differentiate and  \n()  \n(\"  \nthe function which we can EASILY integrate. Recall that our functions are either algebraic  \nfunctions i.e. 􀀫., trig functions, logarithmic functions, exponentials or inverse trig functions. There are many 3 ways you can decide which function to use for  \n􀁖 and which function to use for  \n()  \n.  \n(\"  \n\n| Way 1: Best Method For Beginners Use the acronym LIATE\u003Cbr>Consider all the possible types of functions that can come up:\u003Cbr>\u003Cbr>Locate the two functions given inside in the integral. Whichever of the 2 functions that comes first on the list above is always called 􀁖 .\u003Cbr>For example:\u003Cbr>S 􀀫 / 􀁂􀀯􀀫􀀰􀀫\u003Cbr>􀀫 / is the Algebra function\u003Cb","cbCaienzhOPwVQBy","https://ap.wps.com/l/cbCaienzhOPwVQBy","pdf",514652,"English","# When to use integration by parts\n## Product of unrelated functions\n## Special cases: logarithm and inverse trig\n# Decide between substitution and parts\n## Rule of thumb\n# Core integration by parts workflow\n## Steps 1-4\n# Choosing u and dv\n## LIATE acronym (Way 1)\n## Memorise templates (Way 2)\n## Think ahead strategy (Way 3)","[{\"question\":\"How do you know when integration by parts is the right method?\",\"answer\":\"Use parts when the integral looks like a product of two unrelated functions and substitution or reverse chain rule does not make the integrand easier. If substitution simplifies the integral after rewriting, it should be tried first.\"},{\"question\":\"What are the two harder integration types mentioned, and how should they be handled?\",\"answer\":\"Natural logarithm and inverse trigonometry cases are described as not being direct products and not obvious for parts. They need to be remembered as special cases and forced into a product form to apply integration by parts.\"},{\"question\":\"What is LIATE and how is it used to choose u and dv?\",\"answer\":\"LIATE is an acronym used for beginners to decide which function to label as u and which as dv based on the function types’ order. The function that appears earlier on the LIATE list is chosen as u, and the other becomes the part to integrate accordingly.\"}]","Integration By Parts Cheat Sheet - LIATE | PDF",1791922537]