[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-564853-105":53,"doc-detail-564853-en":120},{"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":113,"head_meta":115,"extra_data":117,"updated_unix":119},105,"en","integral-table-basic-forms-rational-functions","Integral Table - BASIC FORMS - Rational Functions","","Integral Table整理了积分常见类型与对应的基本公式，覆盖基础形式、分式函数、含根号的积分、对数函数、指数函数以及三角函数等多个章节。内容以编号形式给出从简单幂积分到更复杂的组合变形结果，并包含如对数与反三角函数、误差函数 erf 以及三角恒等变换等关键表达，便于在计算与求解时快速查表应用。",{"@graph":63,"@context":112},[64,80,95],{"@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":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/integral-table-basic-forms-rational-functions/564853/",4,{"url":78,"name":59,"@type":81,"author":82,"headline":59,"publisher":85,"fileFormat":88,"inLanguage":57,"description":61,"dateModified":89,"datePublished":89,"encodingFormat":88,"isAccessibleForFree":90,"interactionStatistic":91},"DigitalDocument",{"name":83,"@type":84},"LangkahRina","Person",{"url":68,"name":86,"@type":87},"DocShare","Organization","application/pdf","2026-10-12",true,{"@type":92,"interactionType":93,"userInteractionCount":4},"InteractionCounter",{"@type":94},"ViewAction",{"@type":96,"mainEntity":97},"FAQPage",[98,104,108],{"name":99,"@type":100,"acceptedAnswer":101},"Integral Table包含哪些类型的积分公式？","Question",{"text":102,"@type":103},"文档按章节汇总了基础形式、分式函数、含根号、对数、指数以及多类三角函数相关公式。","Answer",{"name":105,"@type":100,"acceptedAnswer":106},"这些公式如何组织与使用？",{"text":107,"@type":103},"公式以编号形式分组列出，便于在积分计算中按类型快速定位相应结果。",{"name":109,"@type":100,"acceptedAnswer":110},"文档是否包含三角函数与幂/指数组合的积分？",{"text":111,"@type":103},"包含，例如带xn或eax的三角函数积分，以及三角函数幂次与乘积形式的变换结果。","https://schema.org",{"og:url":78,"og:type":114,"og:title":59,"og:site_name":86,"og:description":61},"article",{"robots":116,"canonical":78},"index,follow",{"doc_id":118,"site_id":56},564853,1791844594,{"code":4,"msg":5,"data":121},{"doc_id":118,"user_id":122,"nickname":83,"user_avatar":123,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":124,"file_id":125,"file_url":126,"file_type":127,"file_size":128,"view_count":4,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":79,"language":129,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":130,"faqs":131,"seo_title":132,"seo_description":61,"update_tm":119,"read_time":73},962090893776,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Table of Integrals  \nBASIC FORMS  \n(1) ! xn dx = n~~ ~~~~ ~~1 xn+1  \n(2) ! ~~ ~~1x dx = ln x  \n(3) ! udv = uv\" ! vdu  \n(4) \"u (x)v!(x)dx = u(x)v(x)\\# \"v(x)u!(x)dx  \nRATIONAL FUNCTIONS  \n(5) ! ~~ ~~ax~~ ~~~~ ~~b dx = 1a ln(ax + b)  \n(6) ! ~~ ~~(x~~ ~~+1a)2~~ ~~ dx = xa  \n(7) ! (x + a)n dx = (x + a)n  ~~ ~~1n + 1~~ ~~n~~ ~~  , n ! \"1  \n(8) ! x(x + a)n dx = (x~~ ~~+(an)()xnx1)\"~~ ~~a)  \n(9) ! ~~ ~~1~~ ~~xx2~~ ~~ = tan\"1 x  \n(10) ! ~~ ~~a2~~ ~~x2~~ ~~ = 1a tan\"1(x / a)  \n(11) ! ~~ ~~a2xxx2~~ ~~ = 12 ln(a2 + x2 )  \n(12) ! ~~ ~~ax2xx2~~ ~~ = x \" atan\"1(x / a)  \n(13) ! ~~ ~~ax2xx2~~ ~~ = 12 x2 \" 12 a2 ln(a2 + x2 )  \n(14) \" (ax2 + bx + c)!1dx = 4a2c!~~ ~~b2~~ ~~ tan!1 %\\#$ ~~ ~~~~ ~~24aaxc+!~~ ~~bb~~ ~~&'  \n(15) ! ~~ ~~(x~~ ~~+~~ ~~a1)(x~~ ~~+~~ ~~b) dx = b~~ ~~1\"~~ ~~a [ln(a + x)\" ln(b + x) ], a ! b  \n(16) ! ~~ ~~(x~~ ~~a)2~~ ~~ dx = ~~ ~~a~~ ~~~~ ~~x + ln(a + x)  \n! ~~ ~~ax2~~ ~~+xbx~~ ~~+~~ ~~c dx = ln(ax2~~ ~~+2abx~~ ~~+~~ ~~c)  \n(17) \" ~~ ~~a4bac~~ ~~\"~~ ~~b2~~ ~~ tan\"1 %$\\# ~~ ~~~~ ~~24aaxcbb~~ ~~&'  \nINTEGRALS WITH ROOTS  \n(18)  \n(19)  \n(20)  \n(21)  \n(22)  \n(23)  \n(24)  \n(25)  \n(26)  \n(27)  \n(28)  \n(29)  \n(30)  \n(31)  \n(32)  \n(33)  \n\" x! adx = 23 (x ! a)3/2  \n! ~~ ~~~~ ~~dx = 2x ± a  \n\" ~~ ~~1!~~ ~~dx = ~~ ~~ 2!   \n\"xx! adx = 23 a(x ! a)3/2 + 25 (x ! a)5/2  \n! ax + bdx = $ ~~ ~~2b3a + 2x3~~ ~~ '   \n! (ax + b)3/2 dx = $\"\\# ~~ ~~2b52a~~ ~~ + 4bx5 + 2a5x2~~ ~~ %&  \n! ~~ ~~x±~~ ~~dx = 23(x ± 2a) x ±   \n\" x!~~ ~~dx = !x~~ ~~ ! ! atan!1 \\#$% ~~ ~~xx~~ ~~a!a!~~ ~~x~~  ~~ &'  \n! x~~ ~~~~ ~~adx = x  x + a \"alnx + x + a  \n! xax + bdx = %\\#$\" ~~ ~~415ba22~~ ~~ + 2bx15a + 2x52~~ ~~ &'(   \n\"bx  x 3/2 %  \n! x ax + bdx = \\#$~~ ~~4a~~ ~~ +~~ ~~2~~ ~~&'+ ( b2 ln( 2a x + 2b + ax  )  \n4a3/2  \n! x 3/2 ax~~ ~~+~~ ~~bdx = $%\\# \" b28a2x~~  ~~ + b1x23/2a~~ ~~ + x53/2~~ ~~ &'( +~~ ~~  \n\" b~~ ~~3~~ ~~ln(~~ ~~2a~~ ~~8xa5~~ ~~2b~~ ~~+~~ ~~ax~~  ~~)  \n! ± a2 dx = 12 x±  ± 12 a2 ln(x + x2 ± a2  )  \n\" a2 ! x2 dx = 12 x!  ! 12 a2 tan!1  ~~ ~~xxa22!!ax2~~ ~~&'  \n! x = ~~ ~~13 (x2 ± a2 )3/2  \n! ~~ ~~~~ ~~2~~ ~~ dx = ln(x + x2 ± a2  )  \n©2005 BE Shapiro Page 1  \nThis document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or suitability of this material for any purpose.  \n(34)  \n(35)  \n(36)  \n(37)  \n(38)  \n(39)  \n(40)  \n(41)  \n(42)  \n\" ~~ ~~~~  ~~1!~~  ~~ = sin!1 xa  \n! ~~ ~~x±~~  ~~ =   \n! xax2 + bx + c dx =  \n%$\\# ~~ ~~x33~~ ~~ + 1bx2a + ~~ ~~8ac~~ ~~\"24a32b2~~ ~~ &'( ax2 + bx + c  \n\" b(41ac6a2b2~~ ~~) ln %\\#$ ~~ ~~2ax~~ ~~b + 2ax2 + bc + c&'! ~~ ~~ax2~~ ~~bx~~ ~~+~~ ~~cdx = ~~ ~~1a~~  ~~ ln $\"\\#2ax~~ ~~b + 2ax2 + bx + c%&'  \n! ~~ ~~ax2~~ ~~bx~~ ~~+~~ ~~cdx = 1a ax2 + bx + c  \n\" ~~ ~~2ba3/2~~ ~~ ln%\\#$2ax~~ ~~b + 2ax2 + bx + c  &(  \n'  \nLOGARITHMS  \n(43)  \n(44)  \n(45)  \n(46)  \n(47)  \n(48)  \n! ln xdx = xlnx \" x  \n! ~~ ~~ln(axx) dx = 12 (ln(ax))2  \n! ln(ax + b)dx = ax~~ ~~b ln(ax + b)\" x  \n! ln(a2 x2 ± b2 )dx = x ln(a2 x2 ± b2 ) + 2ba tan\"1  ~~ ~~axb~~ ~~  \" 2x  \n\"ln(a2 ! b2 x2 )dx = x ln(a2 ! b2 x2 ) + 2ab tan!1  ~~ ~~bxa~~ ~~  ! 2x  \n! ln(ax2 + bx + c)dx = 1a 4ac \" b2  tan\"1 %\\#$ ~~ ~~24aaxcbb~~ ~~&'(  \n\"2x +\\#%$ ~~ ~~b2a + x&'(ln(ax2 + bx + c)  \n(49) ! x ln(ax + b)dx = b2a x\" 14~~ ~~ x2 + 12 %\\#$x2 \" b2a2~~ ~~ &'(ln(ax + b)  \n(50) \"x ln(a2 ! b2 x2 )dx = ! ~~ ~~12 x2 + 12 \\#%$x2 ! a2b2~~ ~~ &'(ln(a2 ! bx2 )  \nEXPONENTIALS  \n(51) ! eaxdx = 1a eax  \n(52) ! xeaxdx = 1a xeax + i2a~~  ~~\"3/2~~ ~~ erf(iax  ) where erf (x) = 2~~ ~~ \\#0x e\"t2 dt  \n(53) ! xex dx = (x \" 1)ex  \n(54) ! xeaxdx = % ~~ ~~xa \" a12~~ ~~ &( eax  \n'  \n(55) ! x2 ex dx = ex (x2 \" 2x + 2)  \n(56) ! x2 eaxdx = eax %$\\# ~~ ~~xa2~~ ~~ \" 2ax2 + 2a3~~ ~~ &'  \n(57) ! x 3ex dx = ex (x3 \" 3x2 + 6x \" 6)  \n(58) ! xneaxdx = (\"1)n 1a \\#[1 + n, \"ax] where!(a, x) = $x\\# ta\"1e\"tdt  \n(59) ! eax2 dx = \"i ~~ ~~2\\#a~~  ~~ erf(ixa  )  \nTRIGONOMETRIC FUNCTIONS  \n(60) ! sin xdx = \"cos x  \n(61) ! sin2 xdx = x2 \" 14~~ ~~ sin2x  \n(62) ! sin3 xdx = \" ~~ ~~34 cos x + 112 cos3x  \n(63) ! cos xdx = sin x  \n(64) ! cos2 xdx = x2 + 14~~ ~~","cbCaiq72qTSUi7mL","https://ap.wps.com/l/cbCaiq72qTSUi7mL","pdf",1239546,"English","# BASIC FORMS\n# RATIONAL FUNCTIONS\n# INTEGRALS WITH ROOTS\n# LOGARITHMS\n# EXPONENTIALS\n# TRIGONOMETRIC FUNCTIONS\n# TRIGONOMETRIC FUNCTIONS WITH xn\n# TRIGONOMETRIC FUNCTIONS WITH eax\n# TRIGONOMETRIC FUNCTIONS WITH xn AND eax","[{\"question\":\"Integral Table包含哪些类型的积分公式？\",\"answer\":\"文档按章节汇总了基础形式、分式函数、含根号、对数、指数以及多类三角函数相关公式。\"},{\"question\":\"这些公式如何组织与使用？\",\"answer\":\"公式以编号形式分组列出，便于在积分计算中按类型快速定位相应结果。\"},{\"question\":\"文档是否包含三角函数与幂/指数组合的积分？\",\"answer\":\"包含，例如带xn或eax的三角函数积分，以及三角函数幂次与乘积形式的变换结果。\"}]","Integral Table - BASIC FORMS - Rational Functions | PDF"]