[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572876-105":53,"doc-detail-572876-en":127},{"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":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","calculus-cheat-sheet-integrals-definitions-fundamental-theorem-of-calculus","Calculus Cheat Sheet - Integrals Definitions - Fundamental Theorem of Calculus","","Calculus Cheat Sheet on integrals, organized from core definitions to powerful theorems and practical computation rules. Covers definite integrals via Riemann-sum construction, anti-derivatives and indefinite integrals, and the Fundamental Theorem of Calculus with both parts. Includes integration properties such as linearity, sign changes, constant factors, and symmetry/inequality comparisons. Further provides standard integration techniques: substitution, integration by parts, and guidance for integrating products of trigonometric functions and related trig substitutions.",{"@graph":63,"@context":119},[64,80,102],{"@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/calculus-cheat-sheet-integrals-definitions-fundamental-theorem-of-calculus/572876/",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/calculus-cheat-sheet-integrals-definitions-fundamental-theorem-of-calculus/572876.png","ImageObject",442,249,{"name":88,"@type":89},"Asher","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",6,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"How is the definite integral defined in the cheat sheet?","Question",{"text":109,"@type":110},"The definite integral is defined by dividing [a,b] into n subintervals of width Δx and using a sample point x*i in each interval to form the sum ∑ f(x*i)Δx, giving ∫ from a to b f(x) dx.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"What does the Fundamental Theorem of Calculus state here?",{"text":114,"@type":110},"Part I links g(x)=∫ from a to x f(t)dt to continuity and shows g'(x)=f(x). Part II states that if F is an anti-derivative of f on [a,b], then ∫ from a to b f(x)dx = F(b)-F(a).",{"name":116,"@type":107,"acceptedAnswer":117},"Which techniques does the document list for computing integrals?",{"text":118,"@type":110},"It lists substitution (u-substitution), integration by parts, and methods for products of trigonometric functions. It also includes trig formulas and trig substitutions for integrals involving specific radical expressions.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572876,1792002586,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":47,"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":73},687197207639,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Calculus Cheat Sheet  \nIntegrals Definitions  \nDefinite Integral: Suppose f ( x ) is continuous on [a, b ] . Divide [a, b ] into n subintervals of width D x and choose x*i from each interval. Then Úbaf ( x ) dx =  •Âf ( x*i) D x .  \ni =1  \nAnti-Derivative : An anti-derivative of f ( x ) is a function, F ( x ) , such that F ¢ (x ) = f ( x ) . Indefinite Integral : Ú f ( x ) dx = F ( x ) + c where F ( x ) is an anti-derivative of f ( x ) .  \nFundamental Theorem of Calculus  \nPart I : If f ( x ) is continuous on [a, b ] then g ( x ) = Ú xa f ( t ) dt is also continuous on [a, b ] and g ¢ (x ) = ~~ ~~ddx~~ ~~ Ú xa f ( t ) dt = f ( x ) .  \nPart II : f ( x ) is continuous on [a, b ], F ( x ) isan anti-derivative off ( x ) (i.e. F ( x ) = Ú f ( x ) dx ) then Ú baf ( x ) dx = F ( b )-F ( a ) .  \nVariants of Part I :  \nddx~~ ~~ Ú ua(x ) f ( t ) dt = u ¢ (x ) f u ( x ) ddx~~ ~~ Úvb(x) f ( t ) dt = -v¢ (x ) f v ( x )  \nddx~~ ~~ Úvu()) f ( t ) dt = u ¢ (x ) f [u(x) ]-v¢ (x ) f [v(x) ]  \nProperties  \nÚ f ( x ) ± g ( x ) dx = Ú f ( x ) dx ± Ú g ( x ) dx  \nÚ ba f ( x ) ± g ( x ) dx = Ú ba f ( x ) dx ± Ú ba g ( x ) dx  \nÚ aa f ( x ) dx = 0  \nÚ baf ( x ) dx = - Úba f ( x ) dx  \nÚcf ( x ) dx = c Ú f ( x ) dx , c is a constant  \nÚ ba cf ( x ) dx = c Úba f ( x ) dx , c is a constant  \nÚ ba f ( x ) dx = Ú ba f ( t ) dt  \nÚba f ( x ) dx  \n£ Úba  \nf ( x )  dx  \nIf f ( x ) 􀂳 g ( x ) on a £ x £ b then Úba f ( x ) dx 􀂳 Úba g ( x ) dx  \nIff ( x ) 􀂳 0 on a £ x £ b then Úba f ( x ) dx 􀂳 0  \nIf m £ f ( x ) £ M on a £ x £ b then m( b-a) £ Úbaf ( x ) dx £ M ( b-a )  \nÚ k dx = k x + c  \nÚxn dx = ~~ ~~n1 xn+1 + c, n 􀂹 -1  \nÚx-1 dx = Ú ~~ ~~1x~~ ~~ dx = ln x  + c  \nÚ ~~ ~~ax1+b~~ ~~ dx = ~~ ~~1a~~ ~~ ln ax + b + c  \nÚ ln u du = u ln ( u ) -u + c  \nÚeu du = eu + c  \nCommon Integrals  \nÚcosu du = sin u + c  \nÚsin u du = -cosu + c  \nÚsec2 u du = tan u + c  \nÚsecu tan u du = sec u + c  \nÚcscu cotudu = -cscu + c  \nÚcsc2 u du = -cot u + c  \nÚ tan u du = ln sec u + c  \nÚsec u du = ln sec u + tan u + c Ú ~~ ~~a2~~ ~~u2~~ ~~ du = ~~ ~~1a~~ ~~ tan-1 ( ~~ ~~ua~~ ~~ ) + c  \nÚ ~~ ~~~~  ~~1-~~ ~~2~~  ~~ du = sin-1 ( ~~ ~~ua~~ ~~ ) + c  \nVisit [http://tutorial.math.lamar.edu](http://tutorial.math.lamar.edu) for a complete set of Calculus notes. © 2005 Paul Dawkins  \nCalculus Cheat Sheet  \nStandard Integration Techniques  \nNote that at many schools all but the Substitution Rule tend to be taught in a Calculus II class.  \nu Substitution : The substitution u = g ( x ) will convert Ú ba f ( g ( x ))g ¢ (x ) dx = Úgg((ba)) f ( u ) du using du = g ¢ (x ) dx . For indefinite integrals drop the limits of integration.  \n\n| Ex. Ú 12 5x2 cos ( x3 ) dx\u003Cbr>u = x3 fi du = 3x2 dx fi x2 dx = ~~ ~~13 dux = 1 fi u = 13 = 1 :: x = 2 fi u = 23 = 8 | Ú | 12 5x2 cos ( x3 ) dx = Ú 18 ~~ ~~53 cos ( u ) du\u003Cbr>= ~~ ~~53 sin ( u ) 81 = ~~ ~~53 ( sin ( 8)-sin ( 1)) |\n| --- | --- | --- |\n\nIntegration by Parts : Úu dv = uv- Úv du  \nand Ú bau dv = uv  \nba- Úbav du . Choose u and dv from  \nintegral and compute du by differentiating u and compute v using v = Ú dv .  \nEx. Úxe-x dx  \nu = x dv = e-x fi du = dx v = -e-x Úxe-x dx = -xe-x + Úe-x dx = -xe-x-e-x + c  \n\n| Ex. Ú35 ln x dx\u003Cbr>u = ln x dv = dx fi du = ~~ ~~1x~~ ~~ dx v = x Ú35 ln x dx = x ln x  53- Ú35 dx = ( x ln ( x )-x) | 5\u003Cbr>3 |\n| --- | --- |\n| = 5 ln ( 5)-3ln ( 3)-2 |  |\n\nProducts and (some) Quotients of Trig Functions  \nFor Úsinn x cosm x dx we have the following :  \n1. n odd. Strip 1 sine out and convert rest to cosines using sin2 x = 1-cos2 x , then use the substitution u = cos x .  \n2. m odd. Strip 1 cosine out and convert rest to sines using cos2 x = 1-sin2 x , then use the substitution u = sin x .  \n3. n and m both odd. Use either 1. or 2.  \n4. n and m both even. Use double angle and/or half angle formulas to reduce the integral into a form that can be integrated.  \nFor Ú tan n x secm x dx we have the following :  \n1. n odd. Strip 1 tangent and 1 secant out and convert the rest to secants using  \ntan2 x = sec2 x-1 , then use the substitution u = s","cbCaiuR3NVbhcjfK","https://ap.wps.com/l/cbCaiuR3NVbhcjfK","pdf",187931,"English","# Integrals Definitions\n## Definite Integral and Anti-Derivative\n## Fundamental Theorem of Calculus\n## Properties of Integrals\n# Standard Integration Techniques\n## U-Substitution\n## Integration by Parts\n## Products of Trig Functions and Trig Formulas\n## Trig Substitutions","[{\"question\":\"How is the definite integral defined in the cheat sheet?\",\"answer\":\"The definite integral is defined by dividing [a,b] into n subintervals of width Δx and using a sample point x*i in each interval to form the sum ∑ f(x*i)Δx, giving ∫ from a to b f(x) dx.\"},{\"question\":\"What does the Fundamental Theorem of Calculus state here?\",\"answer\":\"Part I links g(x)=∫ from a to x f(t)dt to continuity and shows g'(x)=f(x). Part II states that if F is an anti-derivative of f on [a,b], then ∫ from a to b f(x)dx = F(b)-F(a).\"},{\"question\":\"Which techniques does the document list for computing integrals?\",\"answer\":\"It lists substitution (u-substitution), integration by parts, and methods for products of trigonometric functions. It also includes trig formulas and trig substitutions for integrals involving specific radical expressions.\"}]","Calculus Cheat Sheet - Integrals Definitions - Fundamental Theorem of Calculus | PDF",1791929817]