[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572420-105":53,"doc-detail-572420-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","derivatives-and-integrals-trig-derivatives-and-integrals-inverse-trig-exponentiallog-and-related-formulas","Derivatives and Integrals - Trig derivatives and integrals, inverse trig, exponential/log, and related formulas","","Midterm formula sheet compiling key results for calculus topics: trigonometric integrals and derivatives, inverse trigonometric derivatives, and standard exponential/log differentiation and integration rules. It also includes hyperbolic-function integral patterns, distance and projection formulas in coordinate geometry, and space-curve tools such as derivative rules, arclength, curvature, TNB (tangent-normal-binormal) frame definitions, and acceleration decomposition. Finally, it provides fundamental trigonometric identities involving sums, differences, double angles, and related transformations.",{"@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/derivatives-and-integrals-trig-derivatives-and-integrals-inverse-trig-exponentiallog-and-related-formulas/572420/",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},"noah","Person",{"url":68,"name":86,"@type":87},"DocShare","Organization","application/pdf","2026-10-13",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},"What are the standard trigonometric integral results included?","Question",{"text":102,"@type":103},"The sheet lists multiple integrals for sin x, cos x, tan x, sec x, cot x, and csc x, plus sec^2 x, csc^2 x, and mixed forms like sec x tan x and csc x cot x.","Answer",{"name":105,"@type":100,"acceptedAnswer":106},"How are inverse trigonometric derivatives presented?",{"text":107,"@type":103},"It provides derivative formulas for arcsin, arccos, arctan, and corresponding csc−1, sec−1, and cot−1 expressions, each written as a simplified algebraic form in x.",{"name":109,"@type":100,"acceptedAnswer":110},"Which geometry and space-curve formulas are included?",{"text":111,"@type":103},"It includes a 3D distance formula, vector projection results, and space-curve formulas for differentiation of parametric vectors, arclength, curvature, the TNB frame, and acceleration decomposition.","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},572420,1791927314,{"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":73,"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":9},1099529839121,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Derivatives and Integrals Trig integrals:  \n1. R sinx dx = 􀀀 cos x + C  \n2. R cosx dx = sin x + C  \n3. R tanx dx = 􀀀 ln j cos xj + C  \n4. R sec x dx = ln j sec x + tan xj + C  \n5. R cotx dx = ln j sin xj + C  \n6. R csc x dx = ln j csc x 􀀀 cot xj + C  \n7. R sec2 x dx = tan x + C  \n8. R csc2 x dx = 􀀀 cotx + C  \n9. R sec x tan x dx = sec x + C  \n10. R cscx cotx dx = 􀀀 cscx + C  \nTrig derivatives:  \n1. ~~d~~dx (sin(x)) = cos(x)  \n2. ~~d~~dx (cos(x)) = 􀀀 sin(x)  \n3. ~~d~~dx (tan(x)) = sec2 (x)  \n4. ~~d~~dx (csc(x)) = 􀀀 csc(x) cot(x)  \n5. ~~d~~dx (sec(x)) = sec(x) tan(x)  \n6. ~~d~~dx (cot(x)) = 􀀀 csc2 (x)  \nInverse trig derivatives:  \n1. ~~d~~dx 􀀀sin􀀀1 (x)􀀁 = p1x~~2~~  \n2. ~~d~~dx (cos􀀀1(x)) = 􀀀 p1x~~2~~  \n3. ~~d~~dx (tan􀀀1(x)) = 1x2  \n4. ~~d~~dx (csc􀀀1(x)) = 􀀀 xp~~1~~x~~2 ~~􀀀1  \n5. ~~d~~dx (sec􀀀1(x)) = xp~~1~~x~~2 ~~􀀀1  \n6. ~~d~~dx (cot􀀀1(x)) = 􀀀 1x2  \nExponential/log:  \n1. ~~d~~dx (bx ) = bx ln b  \n2. R bx dx = l~~b~~nxb + C  \n3. R ekx dx = ~~e~~kkx + C Integrals involving p 1 􀀆 x2  \n1. R p1x~~2~~ dx = sinh􀀀1(x) + C  \n2. R px~~12~~􀀀1 dx = cosh􀀀1(x) + C  \n3. R 1~~ ~~􀀀~~1~~x2 dx = tanh􀀀1(x) + C  \n4. R j~~ ~~xjp~~1~~x~~2 ~~+1 dx = 􀀀 csch􀀀1(x) + C  \n5. R xp1~~1~~􀀀x~~2~~ dx = 􀀀 sech􀀀1(x) + C  \n6. R 1~~ ~~􀀀~~1~~x2 dx = coth􀀀1(x) + C  \n7. R 1~~ ~~􀀀~~1~~x2 dx = ~~1~~2 ln 􀀌 ~~x~~x~~1~~1 􀀌 + C  \n8. R x2~~ ~~1 dx = ~~1~~2 ln 􀀌 ~~x~~x~~1~~1 􀀌 + C  \n9. R px~~12~~+1 dx = ln(x +px2 + 1) + C  \n10. R px~~2~~ dx = ~~x~~2p 1 + x2    \n􀀀 ~~1~~2 ln(x +p 1 + x2 ) + C 11. R xp1~~1~~+x~~2~~ dx = 􀀀 ln 􀀌 p~~x~~2~~ +~~x~~1+1~~ 􀀌 + C 12. R x2~~ ~~px~~2~~ dx = 􀀀 p~~ 1+~~x~~x~~2 + C  \nDistance Formula  \nd((x1 ; y 1 ; z1 ) ; (x2 ; y2 ; z2 )) =p(x1 􀀀 x2 )2 + (y1 􀀀 y2 )2 + (z1 􀀀 z2 )2  \nProjections  \nproja b = ~~a~~a􀀁􀀁~~b~~aa compa b = jproja bj  \nDerivatives/Integrals of Space Curves  \n~~d~~dt hf (t); g (t); h (t)i = h ~~d~~dt f (t) ; ~~d~~dtg (t) ; ~~d~~dt h (t)i  \nRhf(t); g (t); h (t)i dt = hR f (t) dt; R g(t) dt; R h (t) dti  \nArclength  \ns (t) =Rta jr0 (u)j du  \nCurvature  \n􀀔 (t) = 􀀌 ~~d~~dTs 􀀌 = jj~~ ~~d~~d~~ddT~~t~~rt~~ ~~jj = j~~ ~~r0~~ ~~(jtr􀀂(tr)0j~~ ~~0~~3~~(t)j  \nTNB frame  \nT = unit tangent vector, N = j~~ ~~T0T0((tt))~~ ~~j~~ ~~ , B = T 􀀂 N = j~~ ~~r0r0((tt))~~ ~~􀀂􀀂r00r00((tt))~~ ~~j  \nAcceleration  \na (t) = v0 (t)T (t) + 􀀔(t)v (t)2 N (t), where v = j v(t)j  \nTrig Identities  \n1. sin2 􀀒 + cos2 􀀒 = 1  \n2. sin(x 􀀆 y) = sin x cos y 􀀆 cos x sin y  \n3. cos(x 􀀆 y) = cos x cos y 􀀇 sin x sin y  \n4. sin(2􀀒) = 2 sin 􀀒 cos 􀀒  \n5. cos(2􀀒) = cos2 􀀒 􀀀 sin2 􀀒  \n6. tan(2􀀒) = 1~~tan~~tan~~􀀒2~~􀀒  \n7. sin(􀀒=2) = 􀀆 q ~~ 1 ~~􀀀~~ c~~2~~os 􀀒~~  \n8. cos(􀀒=2) = 􀀆 q ~~ 1+c~~2~~os 􀀒~~","cbCaib3ff6vVRaQe","https://ap.wps.com/l/cbCaib3ff6vVRaQe","pdf",126159,"English","# Derivatives and Integrals\n## Trig integrals and trig derivatives\n## Inverse trig derivatives\n## Exponential/log and hyperbolic-function integrals\n## Distance and projections\n## Derivatives/Integrals of space curves, arclength, curvature, TNB frame, acceleration\n## Trig identities","[{\"question\":\"What are the standard trigonometric integral results included?\",\"answer\":\"The sheet lists multiple integrals for sin x, cos x, tan x, sec x, cot x, and csc x, plus sec^2 x, csc^2 x, and mixed forms like sec x tan x and csc x cot x.\"},{\"question\":\"How are inverse trigonometric derivatives presented?\",\"answer\":\"It provides derivative formulas for arcsin, arccos, arctan, and corresponding csc−1, sec−1, and cot−1 expressions, each written as a simplified algebraic form in x.\"},{\"question\":\"Which geometry and space-curve formulas are included?\",\"answer\":\"It includes a 3D distance formula, vector projection results, and space-curve formulas for differentiation of parametric vectors, arclength, curvature, the TNB frame, and acceleration decomposition.\"}]","Derivatives and Integrals - Trig derivatives and integrals, inverse trig, exponential/log, and related formulas | PDF"]