[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-268909-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-268909-en":131},{"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-structured-exam-questions-part-i-evaluate-integrals","Integration Structured Exam Questions Part I - Evaluate integrals","","A set of structured examination questions focused on integration techniques. The tasks require evaluating definite integrals in exact form, including integrals involving trigonometric functions and exponentials, and integrals solvable via substitution and integration by parts. Several questions also require demonstrating identities or equivalences, determining unknown constants from rational-function decompositions, and expressing final results as simplified natural logarithms. Some problems involve parameter determination for positive constants using given numerical constraints.",{"@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/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/integration-structured-exam-questions-part-i-evaluate-integrals/268909/",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-structured-exam-questions-part-i-evaluate-integrals/268909.png","ImageObject",300,407,{"name":42,"@type":43},"Putri","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-24","2026-09-15",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",23,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What does the document ask students to do in general?","Question",{"text":63,"@type":64},"Evaluate specified integrals in exact form, often for definite limits, using appropriate integration methods.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Which integration methods are referenced across the questions?",{"text":68,"@type":64},"The questions explicitly use substitution (e.g., setting u equal to an algebraic expression) and integration by parts, and in one case a trigonometric identity.",{"name":70,"@type":61,"acceptedAnswer":71},"How are some integrals linked to constants or proof steps?",{"text":72,"@type":64},"Several tasks require determining constants from decompositions and then using them to finish an integral, while others ask students to show a stated result holds using a given technique.","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},268909,1789438882,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":96,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":140,"language":141,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":12,"update_tm":80,"read_time":145},962085571259,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Created by T. Madas  \nINTEGRATION STRUCTURED EXAM QUESTIONS PART I  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 1 (**)  \nEvaluate each of the following integrals, giving the answers in exact form.  \na) 􀀁02 ~~ ~~41x+1 dx .  \nπ  \nb) 􀀁π3 cos3x dx .  \n6  \nQuestion 2 (**)  \nBy using the substitution u = 4 + 3x2 , or otherwise, find  \n􀀁 ~~ ~~2x~~ ~~  \ndx .  \n2 2  \n( 4 + 3x )  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 3 (**)  \nShow clearly that  \n1  \n􀀁0 9  \n3 xe3x dx =  1 .  \nQuestion 4 (**)  \n3xx51 ≡ A + ~~ ~~x1 .  \na) Determine the value of each of the constants A and B .  \nb) Hence find  \n􀀁  3x − 5  \ndx .  \nx −1  \nA = 3 , B = −2 ,  \n| 3x − 2 ln | x −1 | + C |\n| --- | --- | --- |\n\nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 5 (**)  \nEvaluate each of the following integrals, giving the answers in exact form.  \na) 􀀁 4 e~~ 1~~2x dx .  \n0  \nπ  \nb) 􀀁04 cos 􀀂􀀄􀀆 3x + ~~ ~~π4~~ ~~ 􀀃􀀇􀀅 dx .  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 6 (**)  \n 5x +13   A   B   \n≡ + .  \n(2x +1)(x + 4 ) 2x +1 x + 4  \na) Determine the value of each of the constants A and B .  \nb) Evaluate  \n􀀁04 ~~ ~~(2xx113x+~~ ~~4~~ ~~) dx ,  \ngiving the answer as a single simplified natural logarithm.  \n C4A  , A = 3 , B = 1 , ln 54  \nQuestion 7 (**)  \nBy using the substitution u 2 = 1 − x 2 , or otherwise, show that  \n􀀁015x ( 1− x2 )~~ 3~~2 dx = 1 .  \nC4A  , proof  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 8 (**)  \nUse integration by parts to find the value of  \n􀀁 π  \n4  \n4x cos 4x dx .  \n0  \nC4A ,  \n−  1  \n2  \nQuestion 9 (**)  \nBy using the substitution u = 1 − x 2 , or otherwise, find  \n􀀁 ~~ ~~12x~~ 3 ~~ dx .  \n( 1− x2 ) 2  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 10 (**)  \nEvaluate each of the following integrals, giving the answers in exact form.  \na) 􀀁03 ~~ ~~2x4+~~ ~~3 dx .  \nπ  \nb) 􀀁06 sin 􀀂􀀄􀀆 4x + ~~ ~~π6~~ ~~ 􀀃􀀇􀀅 dx .  \nC4D , ln 9 ,  \n| \u003Cbr>3 |\n| --- |\n| 4 |\n\nQuestion 11 (**)  \nUse a trigonometric identity to integrate  \n􀀁  1   \ndx . 1+ cos 2x  \n\n| \u003Cbr>, |  ~~ 1~~2 tan x + C  |\n| --- | --- |\n\nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 12 (**)  \n 30   A   B   \n≡ + .  \n(x + 3)(9 − 2x ) x + 3 9 − 2x  \na) Determine the value of each of the constants A and B .  \nb) Evaluate  \n􀀁14 ~~ ~~(x~~ ~~+~~ ~~33)(09~~ ~~−~~ ~~2x~~ ~~) dx ,  \ngiving the answer as a single simplified natural logarithm.  \n C4O  ,  A = 2  ,  B = 4 , 4ln ( ~~ 7~~2) = ln ( ~~ 240~~16~~1~~ )  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 13 (**)  \nEvaluate each of the following integrals, giving the answers in exact form.  \n 1  \n3 −3x  \na) 􀀁0 e dx .  \nπ  \nb) 􀀁04 sin 􀀂􀀄􀀆 2x + ~~ ~~π4~~ ~~ 􀀃􀀇􀀅 dx .  \n| ~~1~~3 ( 1− e−1) | , |  2\u003Cbr>2 |\n| --- | --- | --- |\n\nQuestion 14 (**+)  \nBy using the substitution u 2 = 16 − 7x2 , or otherwise, show that  \n1  \n􀀁  x  dx =  1 .  \n0 16 − 7x2 7  \nC4B  , proof  \nCreated by T. Madas  \nCreated by T. Madas  \nQuestion 15 (**+)  \nDetermine the value of the positive constant k given further that  \n8 4  \n􀀁 ~~ ~~ dx = 1.90038 .  \nk 2x −1  \nGive the value of k to an appropriate degree of accuracy.  \nSYN-P , k ≈ 3.4  \nQuestion 16 (**+)  \nBy using the substitution u = 1+ 4ln x , or otherwise, find  \n􀀁  4   \ndxx (1+ 4ln x )2 .  \nCreated by T. Madas","cbCaiaVhMn7LoUp4","https://ap.wps.com/l/cbCaiaVhMn7LoUp4","pdf",50465791,268,"English","# Question 1\n# Question 2\n# Question 3\n# Question 4\n# Question 5\n# Question 6\n# Question 7\n# Question 8\n# Question 9\n# Question 10\n# Question 11\n# Question 12\n# Question 13\n# Question 14\n# Question 15\n# Question 16","[{\"question\":\"What does the document ask students to do in general?\",\"answer\":\"Evaluate specified integrals in exact form, often for definite limits, using appropriate integration methods.\"},{\"question\":\"Which integration methods are referenced across the questions?\",\"answer\":\"The questions explicitly use substitution (e.g., setting u equal to an algebraic expression) and integration by parts, and in one case a trigonometric identity.\"},{\"question\":\"How are some integrals linked to constants or proof steps?\",\"answer\":\"Several tasks require determining constants from decompositions and then using them to finish an integral, while others ask students to show a stated result holds using a given technique.\"}]","Integration Structured Exam Questions Part I - Evaluate integrals | PDF",675]