[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572936-105":53,"doc-detail-572936-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","strategy-for-integration-four-step-strategy","Strategy for Integration - Four-Step Strategy","","Integration is presented as a strategy-driven process because selecting a method for a given integral is often less direct than choosing a differentiation formula. After recalling basic integration formulas, the text organizes advice into a four-step approach: simplify the integrand, search for an obvious substitution, classify the integrand by form (trigonometric, rational, parts, or radicals), then try again using substitution, integration by parts, algebraic manipulation, or relating to previous problems.",{"@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":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/strategy-for-integration-four-step-strategy/572936/",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/strategy-for-integration-four-step-strategy/572936.png","ImageObject",442,249,{"name":88,"@type":89},"eBook King","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":47},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"Why is integration more challenging than differentiation in this text?","Question",{"text":108,"@type":109},"Integration is described as more challenging because, while differentiation often uses a clear formula, it may not be obvious which technique to apply to a given integral.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What are the four steps in the proposed integration strategy?",{"text":113,"@type":109},"The steps are: (1) simplify the integrand if possible, (2) look for an obvious substitution, (3) classify the integrand by its form, and (4) try again using substitution, parts, manipulation, or connections to previous problems.",{"name":115,"@type":106,"acceptedAnswer":116},"How does the strategy decide between partial fractions, trigonometric substitutions, and integration by parts?",{"text":117,"@type":109},"It uses classification: rational functions suggest partial fractions, products of trigonometric powers suggest trigonometric substitutions, and products involving a polynomial (or power of x) with a transcendental function suggest integration by parts.","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},572936,1792033416,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":47,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":15,"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":79},962088006270,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","Strategy for Integration  \nAs we have seen, integration is more challenging than differentiation. In finding the derivative of a function it is obvious which differentiation formula we should apply. But it may not be obvious which technique we should use to integrate a given function.  \nUntil now individual techniques have been applied in each section. For instance, we usually used substitution in Exercises 5.5, integration by parts in Exercises 5.6, and partial fractions in Exercises 5.7 and Appendix G. But in this section we present a collection of miscellaneous integrals in random order and the main challenge is to recognize which technique or formula to use. No hard and fast rules can be given as to which method applies in a given situation, but we give some advice on strategy that you may find useful. A prerequisite for strategy selection is a knowledge of the basic integration formulas.  \nIn the following table we have collected the integrals from our previous list together with several additional formulas that we have learned in this chapter. Most of them should be memorized. It is useful to know them all, but the ones marked with an asterisk need not be memorized since they are easily derived. Formula 19 can be avoided by using partial fractions, and trigonometric substitutions can be used in place of Formula 20.  \n\n| Table of Integration Formulas Constants of integration have been omitted. |  |  |\n| --- | --- | --- |\n| 1. y xn dx 􀀁 nxn􀀂􀀂~~ ~~11~~ ~~ 􀀅n 􀀂 􀀃1􀀆\u003Cbr>3. y ex dx 􀀁 ex\u003Cbr>5. y sin x dx 􀀁 􀀃 cos x\u003Cbr>7. y sec2x dx 􀀁 tan x\u003Cbr>9. y sec x tan x dx 􀀁 sec x\u003Cbr>11. y sec x dx 􀀁 ln 􀀁 sec x 􀀂 tan x 􀀁\u003Cbr>13. y tan x dx 􀀁 ln 􀀁 sec x 􀀁\u003Cbr>15. y sinh x dx 􀀁 cosh x\u003Cbr>17. y x~~ ~~2~~ ~~dx􀀂~~ ~~a~~ ~~2~~ ~~ 􀀁 ~~ ~~1a~~ ~~ tan􀀃1􀀃 ~~ ~~xa~~ ~~ 􀀄\u003Cbr>*19. y x~~ ~~2~~ ~~~~ ~~a~~ ~~2~~ ~~ 􀀁 ~~ ~~21a~~ ~~ ln 􀀂 ~~ ~~xx~~ ~~~~ ~~aa~~ ~~ 􀀂 | 2. y\u003Cbr>4. y\u003Cbr>6. y\u003Cbr>8. y\u003Cbr>10. y\u003Cbr>12. y\u003Cbr>14. y\u003Cbr>16. y\u003Cbr>18. y\u003Cbr>*20. y | ~~ ~~1x~~ ~~ dx 􀀁 ln 􀀁 x 􀀁 ax dx 􀀁 ~~ ~~lanxa cos x dx 􀀁 sin x\u003Cbr>csc2x dx 􀀁 􀀃cot x\u003Cbr>csc x cot x dx 􀀁 􀀃 csc x csc x dx 􀀁 ln 􀀁 csc x 􀀃 cot x 􀀁 cot x dx 􀀁 ln 􀀁 sin x 􀀁\u003Cbr>cosh x dx 􀀁 sinh x\u003Cbr>s~~a ~~d2~~ x ~~2~~ ~~ 􀀁 sin􀀃1􀀃 ~~ ~~xa~~ ~~ 􀀄 s~~x ~~2dx􀀁~~ a ~~2~~ ~~ 􀀁 ln 􀀁 x 􀀂 sx 2 􀀁 a2 􀀁 |\n\n2 ■ STRATEGY FOR INTEGRATION  \nOnce you are armed with these basic integration formulas, if you don’t immediately see how to attack a given integral, you might try the following four-step strategy.  \n1. Simplify the Integrand if Possible Sometimes the use of algebraic manipulation or trigonometric identities will simplify the integrand and make the method of integration obvious. Here are some examples:  \ny sx (1 􀀂 sx ) dx 􀀁 y (sx 􀀂 x) dxy ~~ ~~ae2􀀅􀀅~~ ~~ d􀀅 􀀁 y ~~ ~~sc~~ ~~􀀅􀀅~~ ~~ cos2 􀀅 d􀀅  \n􀀁 y sin 􀀅 cos 􀀅 d􀀅 􀀁 1~~2~~ y sin 2􀀅 d􀀅 y 􀀁sin x 􀀂 cos x􀀂2 dx 􀀁 y 􀀁sin2x 􀀂 2 sin x cos x 􀀂 cos2x􀀂 dx  \n􀀁 y 􀀁1 􀀂 2 sin x cos x􀀂 dx  \n2. Look for an Obvious Substitution Try to find some function u 􀀁 t􀀁x􀀂 in the integrand whose differential du 􀀁 t 􀀄􀀁x􀀂 dx also occurs, apart from a constant factor. For instance, in the integral  \ny ~~ ~~x~~ ~~2~~ ~~~~ ~~1~~ ~~ dx  \nwe notice that if u 􀀁 x 2 􀀁 1, then du 􀀁 2x dx. Therefore, we use the substitution u 􀀁 x 2 􀀁 1 instead of the method of partial fractions.  \n3. Classify the Integrand According to Its Form If Steps 1 and 2 have not led to the solution, then we take a look at the form of the integrand f 􀀁x􀀂 .  \n(a) Trigonometric functions. If f 􀀁x􀀂 is a product of powers of sin x and cos x, of tan x and sec x, or of cot x and csc x, then we use the substitutions recommended in Section 5.7 and Additional Topics: Trigonometric Integrals.  \n(b) Rational functions. If f is a rational function, we use the procedure involving partial fractions in Section 5.7 and Appendix G.  \n(c) Integration by parts. If f 􀀁x􀀂 is a product of a power of x (or a polynomial) and a transcendental function (such as a trigonometric, exponential, or logarithmic function), then we try integration by parts, choosing u and dv accordin","cbCaivDK5zGkjVXz","https://ap.wps.com/l/cbCaivDK5zGkjVXz","pdf",501139,"English","# Strategy for Integration\n## Four-step strategy for solving integrals\n## Basic integration formulas and common techniques","[{\"question\":\"Why is integration more challenging than differentiation in this text?\",\"answer\":\"Integration is described as more challenging because, while differentiation often uses a clear formula, it may not be obvious which technique to apply to a given integral.\"},{\"question\":\"What are the four steps in the proposed integration strategy?\",\"answer\":\"The steps are: (1) simplify the integrand if possible, (2) look for an obvious substitution, (3) classify the integrand by its form, and (4) try again using substitution, parts, manipulation, or connections to previous problems.\"},{\"question\":\"How does the strategy decide between partial fractions, trigonometric substitutions, and integration by parts?\",\"answer\":\"It uses classification: rational functions suggest partial fractions, products of trigonometric powers suggest trigonometric substitutions, and products involving a polynomial (or power of x) with a transcendental function suggest integration by parts.\"}]","Strategy for Integration - Four-Step Strategy | PDF",1791930163]