[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572933-105":53,"doc-detail-572933-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","integration-formulas-math-integration-guide","Integration Formulas - Math Integration Guide","","Integration Formulas compiles a structured reference of indefinite integral techniques and results. It covers common integrals using substitution and integration by parts, then organizes more advanced families: rational functions, trigonometric functions, exponential functions, and logarithmic functions. The content provides standard antiderivatives, conditional forms based on parameters, and typical transformations that support symbolic integration in calculus.",{"@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/integration-formulas-math-integration-guide/572933/",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/integration-formulas-math-integration-guide/572933.png","ImageObject",442,249,{"name":88,"@type":89},"Stanford","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-20","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},"What two main methods are introduced for indefinite integration?","Question",{"text":108,"@type":109},"The document presents the method of substitution and integration by parts as primary techniques for computing indefinite integrals.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How are integrals of rational and irrational functions organized?",{"text":113,"@type":109},"They are grouped into general formula patterns, including cases for powers of x and forms involving x with parameter-based substitutions, often adding a constant of integration.",{"name":115,"@type":106,"acceptedAnswer":116},"Which categories of special function integrals are included after rational functions?",{"text":117,"@type":109},"It includes integrals of trigonometric functions, exponential functions, and logarithmic functions, each with common antiderivative forms and related transformations.","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},572933,1792022859,{"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":47,"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":73},2336477552062,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","[www.mathportal.org](www.mathportal.org)  \nIntegration Formulas  \n1. Common Integrals  \nIndefinite Integral  \nMethod of substitution ∫ f (g(x))g ′(x)dx = ∫ f (u)du Integration by parts  \n∫ f (x) g ′(x)dx = f (x) g (x) − ∫ g (x)f ′(x)dx  \nIntegrals of Rational and Irrational Functions  \n∫ xn dx = ~~ ~~xn11+ C ∫ ~~ ~~1xdx = ln x  + C  \n∫ cdx = cx + C ∫ xdx = ~~ ~~x22~~ ~~ + C  \n∫ x2dx = ~~ ~~x33~~ ~~ + C  \n∫ ~~ ~~x12~~ ~~dx = − ~~ ~~1x + C  \n∫ xdx = ~~ ~~2x3x~~  ~~ + C ∫ ~~ ~~1~~ ~~+1x2~~ ~~dx = arctan x + C ∫ ~~ ~~1~~ ~~x~~ ~~dx = arcsin x + C  \nIntegrals of Trigonometric Functions  \n∫ sin x dx = − cos x + C ∫ cos x dx = sin x + C  \n∫ tan x dx = ln sec x + C ∫ sec x dx = ln tan x + sec x  + C ∫ sin 2 x dx = ~~ ~~12 (x − sin x cos x ) + C ∫ cos2 x dx = ~~ ~~12 (x + sin x cos x ) + C ∫ tan 2 x dx = tan x − x + C ∫ sec2 x dx = tan x + C  \nIntegrals of Exponential and Logarithmic Functions  \n∫ ln x dx = xlnx − x + C  \n∫ xn ln xdx = xn11 ln x − ~~ ~~(~~ ~~nx+11)2~~ ~~ + C ∫ ex dx = ex + C  \n∫ bx dx = lbnxb + C ∫ sinhx dx = coshx + C ∫ coshx dx = sinhx + C  \n[www.mathportal.org](www.mathportal.org)  \n2. Integrals of Rational Functions  \nIntegrals involving ax + b  \n∫ (ax + b )n dx = (~~ ~~axa(+n)n11~~ ~~ ( for n ≠ −1)∫ ~~ ~~ax~~ ~~~~ ~~b~~ ~~ dx = 1a~~ ~~ ln ax + b  \n∫ x (ax + b )n dx = a~~ ~~a2((nn11))x(nb2~~ ~~) (ax + b )n+1 ( for n ≠ −1 , n ≠ −2 )  \n∫ ~~ ~~ax~~ ~~b dx = xa − ~~ ~~ba2~~ ~~ ln ax + b ∫  x  dx =  b  +  1 ln ax + b (ax + b )2 a 2 (ax + b ) a 2  \n∫ ~~ ~~(ax~~ ~~~~ ~~b~~ ~~)n~~ ~~ dx = a~~ ~~2~~ ~~(n~~ ~~−~~ ~~a1)(1(2)~~ ~~x)~~ ~~(bx+~~ ~~b~~ ~~)n−1~~ ~~ ( for n ≠ −1, n ≠ −2 )∫ ~~ ~~axx~~ ~~b~~ ~~ dx = a13~~ ~~ 􀀆􀀄􀀂􀀄 (ax~~ ~~+2~~ ~~b~~ ~~)2~~ ~~ − 2b (ax + b ) + b2 ln ax + b􀀃􀀇􀀅  \n∫ ~~ ~~(axb~~ ~~)2~~ ~~ dx = a13~~ ~~ 􀀄􀀂􀀆ax + b − 2b ln ax + b − ~~ ~~abx~~ ~~b~~ ~~ 􀀃􀀇􀀅  \n2 􀀂 2 􀀃∫  x  dx =  1  􀀄 ln ax + b +  2b  −  b  􀀅(ax + b )3 a3 􀀆􀀄 ax + b 2 (ax + b )2 􀀇􀀅  \n∫ ~~ ~~(axb~~ ~~)n~~ ~~ dx = a13~~ ~~ 􀀆􀀄􀀄􀀂 − ~~ ~~(axn+−b3)3−n~~ ~~ + ~~ ~~2b~~ ~~(~~ ~~anb2)2−n~~ ~~ − ~~ ~~b2~~ ~~(~~ ~~axnb1)1−n~~ ~~􀀃􀀇􀀅 ( for n ≠ 1, 2,3 )∫ ~~ ~~x~~ ~~(a1x+~~ ~~b~~ ~~) dx = − ~~ ~~1b~~ ~~ ln  ~~ ~~ax~~ ~~b~~ ~~   \n∫  1  dx = − 1  +  a  ln ax + b  \nx2 (ax + b ) bx b2 x  \n 1  􀀂 1 1 2 ax + b􀀃  \n∫ x2 (ax + b )2 dx = −a 􀀆􀀄 b2~~ ~~(a~~ ~~+~~ ~~xb~~ ~~) +  ab2 x − b3  ln   x  􀀇􀀅  \nIntegrals involving ax2 + bx + c  \n∫ ~~ ~~x2~~ ~~~~ ~~a~~ ~~2~~ ~~ dx = 1a~~ ~~ arctg ~~ ~~xa  \n􀀈 1 a − x ∫ ~~ ~~x2~~ ~~~~ ~~a~~ ~~2~~ ~~ dx = 􀀊  21a lnalnx  xa  for x  \u003C a   \n􀀉􀀋2a x + a for x > a  \n[www.mathportal.org](www.mathportal.org)  \n􀀈 2 2ax + b  \n􀀉  2  arctan  2   \n􀀉  4ac − b  4ac − b  \n1 􀀉 2 2ax + b − b − 4ac  \n∫  ax2 + bx + c dx = 􀀊􀀉 b − 4ac  ln   2ax + b + b − 4ac 􀀉  \n􀀉 −  2   \n􀀋􀀉 2ax + b  \n∫ ~~ ~~ax2~~ ~~+xbx~~ ~~+~~ ~~c dx = ~~ ~~21a~~ ~~ ln ax2 + bx + c  − ~~ ~~b2a~~ ~~ ∫ ~~ ~~ax2~~ ~~xbx~~ ~~+~~ ~~c  \nfor 4ac − b2 > 0  \nfor 4ac − b2 \u003C 0  \nfor 4ac − b2 = 0  \n􀀈 m 2  \n􀀉 ~~ ~~ ln  ax  \n􀀉 2a  \nmx + n 􀀉 m 2  \n∫  ax2 + bx + c  dx = 􀀊􀀉 2a  ln ax  \n􀀉  \n􀀉  m  ln ax2  \n􀀉􀀋2a  \n2an − bm 2ax + b 2  \n+ bx + c  +  a4ac − b arctan  4ac − b  for 4ac − b > 0  2an − bm   2ax + b  2  \n+ bx + c  + arctanh for 4ac − b \u003C 0 ab − 4ac b − 4ac  \n 2an − bm  2  \n+ bx + c  − for 4ac − b = 0 a (2ax + b )  \n∫ ~~ ~~(~~ ~~ax2~~ ~~+~~ ~~1bx~~ ~~+~~ ~~c~~ ~~)n~~ ~~ dx = ~~ ~~(n~~ ~~−1)~~ ~~(~~ ~~4ac~~ ~~−~~ ~~2ab2x)~~ ~~bx2~~ ~~+~~ ~~bx~~ ~~+~~ ~~c)n−1~~ ~~ + ~~ ~~(n~~ ~~−(~~ ~~21n)~~ ~~(3ac)~~ ~~b2~~ ~~)~~ ~~ ∫ ~~ ~~(~~ ~~ax2~~ ~~+~~ ~~b1x~~ ~~+~~ ~~c)n−1~~ ~~ dx ∫ ~~ ~~x~~ ~~(~~ ~~ax2~~ ~~bx~~ ~~+~~ ~~c~~ ~~) dx = ~~ ~~21c~~ ~~ ln  ~~ ~~ax2~~ ~~2bx~~ ~~+~~ ~~c~~ ~~  − ~~ ~~b2c∫ ~~ ~~ax2~~ ~~+1bx~~ ~~+~~ ~~c dx  \n3. Integrals of Exponential Functions  \n∫ xecxdx = eccx2~~ ~~ (cx − 1)∫ x2 ecxdx = ecx 􀀂􀀄􀀆 ~~ ~~x2c~~ ~~ − ~~ ~~2xc2 + ~~ ~~2c3~~ ~~􀀃􀀇􀀅∫ xnecxdx = 1c xnecx − ~~ ~~nc~~ ~~ ∫ xn−1ecxdx ∫  ecx  dx = ln x +   (cx )i x i=1 i ⋅ i!  \n∫ ecx lnxdx = ~~ ~~1c ecx ln x + Ei (cx )  \n∫ ecx sin bxdx = ~~ ~~c~~ ~~2eb2~~ ~~ (csinbx − bcosbx )  \n∫ ecx cos bxdx = c~~ ~~2eb2~~ ~~ (ccosbx + bsin bx )  \n∫ ecx ","cbCaihyI1QoykXDz","https://ap.wps.com/l/cbCaihyI1QoykXDz","pdf",81167,"English","# Common Integrals\n## Indefinite Integral\n## Integrals of Rational and Irrational Functions\n## Integrals of Trigonometric Functions\n## Integrals of Exponential and Logarithmic Functions\n# Integrals of Rational Functions\n## Integrals involving ax + b\n## Integrals involving ax^2 + bx + c\n# Integrals of Exponential Functions\n# Integrals of Logarithmic Functions\n# Integrals of Trig. Functions","[{\"question\":\"What two main methods are introduced for indefinite integration?\",\"answer\":\"The document presents the method of substitution and integration by parts as primary techniques for computing indefinite integrals.\"},{\"question\":\"How are integrals of rational and irrational functions organized?\",\"answer\":\"They are grouped into general formula patterns, including cases for powers of x and forms involving x with parameter-based substitutions, often adding a constant of integration.\"},{\"question\":\"Which categories of special function integrals are included after rational functions?\",\"answer\":\"It includes integrals of trigonometric functions, exponential functions, and logarithmic functions, each with common antiderivative forms and related transformations.\"}]","Integration Formulas - Math Integration Guide | PDF",1791930156]