[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573942-105":53,"doc-detail-573942-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","table-of-useful-integrals","Table of Useful Integrals","","A reference collection of mathematical results centered on useful integrals and related analytic formulas. The content includes definite and indefinite integration identities, common transformations such as integration by parts, and standard series expansions including Taylor and geometric series. It also provides selected advanced formulae for Euler’s formula, quadratic and polynomial solutions, differential equation solutions with constant coefficients, coordinate conversions in spherical polar coordinates, commutator identities, and creation/annihilation operator relations, plus atomic units placeholders.",{"@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":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/table-of-useful-integrals/573942/",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/table-of-useful-integrals/573942.png","ImageObject",442,249,{"name":88,"@type":89},"Riley West","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-14",true,{"@type":98,"interactionType":99,"userInteractionCount":76},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What types of integral formulas are included in the table?","Question",{"text":108,"@type":109},"The document provides multiple integral identities, including definite/indefinite forms and expressions involving exponential, sine, and cosine terms. It also lists results related to products of trigonometric functions and powers.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Does the table include series expansions?",{"text":113,"@type":109},"Yes. It contains formulas for Taylor Series and Geometric Series in summation form.",{"name":115,"@type":106,"acceptedAnswer":116},"What other mathematical topics appear besides integrals?",{"text":117,"@type":109},"It includes Euler’s formula, solutions related to quadratic and higher-order polynomials, a constant-coefficient second-order homogeneous differential equation form, spherical polar to Cartesian coordinate conversions, and commutator/operator identities. It also mentions atomic units and physical constants as sections.","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},573942,1792011663,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":76,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"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},1099523885074,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Table of Useful Integrals, etc.  \n∞ 2 1 􀀣 π 􀀦 e− ax dx = 2 􀀥􀀤 ~~ ~~a~~ ~~ 􀀧􀀨  \n∞ 2 − ax2 1 􀀣 x e dx = 4a 􀀥􀀤  \n12  \nπ  􀀦  \na 􀀧􀀨  \n12  \nx 2ne − ax2 dx = 1⋅~~ ~~3~~ ~~⋅~~ ~~52⋅n⋅+⋅1nn~~ ~~−~~ ~~1~~) ~~0  \n∞ n!  \n∫x ne− ax dx = an+1 0  \n􀀤 π  􀀧􀀦􀀥 a 􀀨􀀩  \n12  \n∞∫  \n0  \n∞∫  \n0  \n∞∫  \n0  \n2 1 xe − ax dx = 2a  \nx 3e − ax2 dx = 21a2 x 2n+1e − ax2 dx = 2nan!+1  \nIntegration by Parts:  \nb b  \n∫UdV = UVba − ∫VdU U and V are functions of x. Integrate from x = a to x = b  \na a  \n∫sin (ax ) dx = − ~~ ~~1a cos( ax )  \n∫sin2 ( ax ) dx = x2 − sin4~~( ~~2aax~~ )~~  \n∫sin3 ( ax ) dx = − ~~ ~~1a cos( ax ) + 31a cos3 ( ax )  \n∫sin4 ( ax )dx = 3x8 − 3sin16~~( ~~2aax~~ )~~ − sin3~~ (~~ax4~~)~~caos~~ ( ~~ax~~ )~~  \n∫sin (ax )sin (bx ) dx = sin~~(~~(~~ ~~aa~~ ~~−~~ ~~bb~~)~~)~~ ~~x~~ ~~~~ ~~ − sin~~(~~(~~ ~~aa~~ ~~~~ ~~bb~~)~~)~~ ~~x~~ ~~~~ ~~ where a2 ≠ b2  \n∫cos( ax )cos( bx ) dx = sin~~(~~(~~ ~~aa~~ ~~−~~ ~~bb~~)~~)~~ ~~x~~ ~~~~ ~~ + sin~~(~~(~~ ~~aa~~ ~~~~ ~~bb~~)~~)~~ ~~x~~ ~~  \n∫sin (ax )cos( bx ) dx = −~~ ~~cos2􀀢~~􀀣(~~a(~~ ~~a−−bb~~) ~~)x~~ ~~~~ ~~ − co~~(~~(aa~~ ~~~~ ~~bb~~)~~)~~ ~~x~~ ~~  \n∫x sin2 ( ax ) dx = x42~~ ~~ − x~~ ~~sin4~~(~~2aax~~ )~~ − cos8~~(~~2a2ax~~ )~~  \n∫x2 sin2 ( ax ) dx = x63 − 􀀢􀀣􀀤 x24a − 81a3~~ ~~ 􀀥􀀦􀀧 sin (2ax) − x~~ ~~cos4a~~( ~~22ax~~ )~~  \n∫x sin (ax )sin (bx ) dx = cos2(−−bb)x~~ ~~~~ ~~ − cos2(++bb)x~~ ~~~~ ~~ + x~~ ~~sin2(a(~~ ~~b)2)x~~ ~~~~ ~~ − x~~ ~~sin2(a(~~ ~~b)2)x~~ ~~  \n∫x sin (ax ) dx = sina~~(~~ax2~~)~~ − x~~ ~~cosa~~( ~~ax~~ )~~  \n∫x cos( ax ) dx = x~~ ~~sina~~( ~~ax~~ )~~ + cosa~~(~~2ax~~ )~~  \n∫cos( ax ) dx = sina~~( ~~ax~~ )~~  \n∫cos2 ( ax ) dx = x2 + sin4~~( ~~2aax~~ )~~  \n∫x2 cos2 ( ax ) dx = x63 + 􀀢􀀣􀀤 x24a − 81a3~~ ~~ 􀀥􀀦􀀧 sin (2ax) + x~~ ~~cos4a~~( ~~22ax~~ )~~  \n∫cos( bx )e − ax2 dx = ~~(~~a2eb2~~ )~~  acos( bx ) + bsin (bx )   \nTaylor Series:  \n∑ nn∞0 ~~ ~~f~~ ~~(nn)~~ (~~!xo~~ )~~ (x − xo )n  \nGeometric Series:  \n∞ 1  \n0 xn = 1~~ ~~−~~ ~~x  \n\n| Euler’s Formula:\u003Cbr>eiφ = cosφ + isinφ |\n| --- |\n| Quadratic Equation and other higher order polynomials:\u003Cbr>ax 2 + bx + c = 0\u003Cbr>−b ± b2 − 4ac x =\u003Cbr>2a\u003Cbr>ax 4 + bx2 + c = 0\u003Cbr>􀀢 −b ± b2 − 4ac 􀀥 x = ± 􀀣􀀤 ~~ ~~2a~~ ~~ 􀀦􀀧 |\n| General Solution for a Second Order Homogeneous Differential Equation with Constant Coefficients:\u003Cbr>If: y ʹ′ʹ′ + py ʹ′ + qy = 0\u003Cbr>Assume a solution for y:\u003Cbr>y = esx y ʹ′ = sesx y ʹ′ʹ′ = s2 esx\u003Cbr>∴ s2 esx + psesx + qesx = 0 and s2 + ps + q = 0 Hence y = c1es1x + c2 es2x |\n| Conversions from spherical polar coordinates into Cartesian coordinates:\u003Cbr>x = r sinθ cosφ y = r sinθ sinφ x = r cosθ\u003Cbr>dv = r2 sinθdrdθdφ 0 \u003C r \u003C ∞\u003Cbr>0 \u003C θ \u003C π\u003Cbr>0 \u003C φ \u003C 2π |\n\nCommutator Identities:  \n ˆA, ˆB  = −  ˆB, ˆA  \n ˆA, ˆAn  = 0 n = 1, 2, 3, 􀀡  kˆA, ˆB  =  ˆA, kˆB  = k  ˆA, ˆB   ˆA + ˆB, ˆC  =  ˆA, ˆC  +  ˆB, ˆC   ˆA, ˆBˆC  =  ˆA, ˆB  ˆC + ˆB  ˆA, ˆC   \n ˆAˆB, ˆC  =  ˆA, ˆC  ˆB + ˆA  ˆB, ˆC   \nCreation and Annihilation Operators  \nl±  l , ml , s , ms  = ( l ( l + 1) − ml ( ml ± 1))2 􀀡  l , ml ± 1 , s , ms s±  l , ml , s , ms  = (s (s + 1) − ms ( ms ± 1))2 􀀡  l , ml , s , ms ±1  j±  j , mj  = ( j ( j + 1) − mj ( mj ± 1))2 􀀡  j , mj ±1   \nAtomic Units:  \nPhysical Constants:  \nOperators:","cbCaiqZIF0XhQPIl","https://ap.wps.com/l/cbCaiqZIF0XhQPIl","pdf",2345771,6,"English","# Useful Integrals\n## Integration identities and by-parts results\n## Series expansions and core formulas\n## Differential equations and coordinate conversions\n## Commutator and operator identities","[{\"question\":\"What types of integral formulas are included in the table?\",\"answer\":\"The document provides multiple integral identities, including definite/indefinite forms and expressions involving exponential, sine, and cosine terms. It also lists results related to products of trigonometric functions and powers.\"},{\"question\":\"Does the table include series expansions?\",\"answer\":\"Yes. It contains formulas for Taylor Series and Geometric Series in summation form.\"},{\"question\":\"What other mathematical topics appear besides integrals?\",\"answer\":\"It includes Euler’s formula, solutions related to quadratic and higher-order polynomials, a constant-coefficient second-order homogeneous differential equation form, spherical polar to Cartesian coordinate conversions, and commutator/operator identities. It also mentions atomic units and physical constants as sections.\"}]","Table of Useful Integrals | PDF",1791954480]