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ÊÁѤÃÊÁÒªÔ¡ | ¤ÙèÁ×Í¡ÒÃãªé | ÃÒª×èÍÊÁÒªÔ¡ | »¯Ô·Ô¹ | ¢éͤÇÒÁÇѹ¹Õé | ¤é¹ËÒ |
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à¤Ã×èͧÁ×ͧ͢ËÑÇ¢éÍ | ¤é¹ËÒã¹ËÑÇ¢é͹Õé |
#1
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integral ¨Ó¡Ñ´à¢µ¢é͹Õé·Ó䧤ÃѺ
⨷Âì¤×Í integral µÑé§áµè 0 ¶Ö§ pi
¢Í§ (x sinx)/(1 + (cosx)^2) ------------- ÍÂÒ¡ä´éÊͧÇÔ¸Õ¤×Í 1) ËÒ integral äÁè¨Ó¡Ñ´à¢µ¡è͹ 2) ¨Ñ´ÃÙ»ãËéàËÁÒÐÊÁáÅéǵͺä´éàÅ (àËç¹à¤éҺ͡ÇèÒ·Óä´é 2 ÇÔ¸Õ) ªèÇÂ˹è͹ФÃѺ ÍèÒ¹ÁÒ¶Ö§àÃ×èͧ¹ÕéáÅéÇ·ÓäÁè ä´é àÅÂäÁè¡ÅéÒÍèÒ¹µèÍ |
#2
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xlover13 àÃÕ¹ªÑé¹ä˹¤ÃѺ
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#3
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¼Á¨º Á.6 áÅéǤÃѺ¤Ø³ Meijin ¡ÓÅѧÃͼÅ
Ent ÍÂÙè µÍ¹¹ÕéÅͧÍèÒ¹ cal Åèǧ˹éÒ´Ù ¡çàÅÂà¨Í»ÑËÒÍÂèÒ§·ÕèÇèÒ¹ÕèáËÅФÃѺ àÍ.. ¼ÁÇèÒ¼Áà¨Í¤Ø³ã¹ dek-d.com ÃÖà»ÅèÒ¤ÃѺ (àÍÒà»ç¹ÇèÒã¤ÃµÍºä´éªèǼÁ´éÇ ¼ÁÍÂÒ¡ÃÙé ÇÔ¸Õ·ÓÁÒ¡ æ àŤÃѺ) |
#4
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¢é͹Õé äÁèÃÙéÇèÒ¾Õè¤è͹¢éÒ§¨Ðà«èÍ¡ÇèÒ·Õè¹éͧá¹ÐÁÒã¹ ¢éÍ 2. ÇèҨѴä´é§èÒ æ áÅéǵͺàÅÂÃÖà»ÅèÒ
µÍº 4pi / 3 (µÑÇàÅ¢·Õèá·¹ÍÒ¨¤Ó¹Ç³¼Ô´ä´é áµèÇÔ¸Õ¡ÒäԴÇèÒäÁè¼Ô´) ÅͧÁÒ´ÙÇÔ¸Õ·Õè¾Õè¤Ô´¹Ð¤ÃѺ. ËÅÑ¡¡Òâé͹Õé ¤×Í ãªé by part Åéǹ æ áµè·Õè¹ÕéÇÔ¸Õ¡Òà by part ·ÕèÇèÒ ¤×Í integrate( u dv ) = uv - integrate( v du ) ¹Ñé¹ àÃÒµéͧ·Ó¡ÒÃÍÔ¹·Ôà¡Ãµ 2 ÍÂèÒ§·ÕèÇèÒµèÍ仹ÕéãËéáÁè¹ÂÓ¡è͹ ¤×Í 1. integrate(cos3x) = (1/3) sin3x + C à»ç¹µé¹ 2. integrate (x sinx) dx = sinx - x cosx + C ( â´Â by part àÅ×Í¡ u = x áÅÐ dv = x sinx dx ) à»ç¹µé¹ ¨Ðä´éµÒÁ⨷ÂìÇèÒ (x sinx)/(1 + (cosx)^2) dx àÅ×Í¡ u = 1 + (cos x )^2 áÅÐ dv = x sinx dx ¨Ðä´éÇèÒÁѹ¤×Í [1+ (cos x )^2 ] [ sin x - x cosx ] - integrate[ 2 (sin x - x cos x)sin x cos x dx ] ...(1) ¾Ô¨ÒóҵÑÇËÅѧ ¤×Í integrate[ 2 (sin x - x cos x)sin x cos x dx ] ¡ÃШÒÂÍÍ¡ÁÒà»ç¹ 2 ¾¨¹ì ¤×Í ¾¨¹ì·Õè 1 . integrate [ 2 (sin x) ^2 cos x dx ] ¾¨¹ì·Õè 2. integrate [ 2 x sin x (cos x)^2 dx ] ¾¹¨ì·Õè 1 ¨Ðä´é¼ÅÅѾ¸ì ¤×Í (1/2) [ sin x - (1/3) sin 3x ] ....(2) ¾¨¹ì·Õè 2. by part â´ÂàÅ×Í¡ u = 2x áÅÐ dv = sin x (cos x )^2 dx ¨Ðä´é¼ÅÅѾ¸ì ¤×Í (-x/2)[cos x + (1/3) cos 3x ] - (1/2) [ sin x + (1/9) sin 3x ] ....(3) ÊØ´·éÒÂá·¹¤èÒµèÒ§ æ ¨Ò¡ (2) áÅÐ (3) ŧä»ã¹ (1) ¨Ðä´é ·Õèµéͧ¡Òà áÅéÇá·¹¤èÒ x = 0 ¶Ö§ pi «Ö觨Ðä´éà»ç¹ 4pi/3 - 0 = 4pi / 3 note. ã¹áµèÅТÑé¹ÍÒ¨µéͧ·Ó¡Òà integrate â´Â bypart ÍÕ¡·Õ â´Â੾Òоǡ·ÕèÁÕ x ¤Ù³ÍÂÙè ãËéàÅ×Í¡ u = x ËÃ×Í 2x à»ç¹µé¹ àÊÁÍ. ⨷Âì¢é͹ÕéµÒÁ¤ÇÒÁ¤Ô´¢Í§¾Õè ¼Ùé·ÕèàÃÔèÁÈÖ¡ÉÒÂѧäÁè¤ÇèзӹРà¾ÃÒÐÍÒ¨»Ç´ËÑÇÁÒ¡ä»Ë¹èÍÂ. áµèÍÂèÒ§·Õè¹éͧá¹ÐÁÒã¹¢éÍ 2. ÍÒ¨ÁÕÇÔ¸Õ·Õè©ÅÒ´¡ÇèÒ¹Õé 10 à·èÒ ¡çä´é¤ÃѺ. |
#5
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¾Õè¡Ã¤ÃѺ ¢Íº¤Ø³ÁÒ¡·ÕèáÊ´§ãËé´Ù
¤ÓµÍºã¹Ë¹Ñ§Ê×Í·Õè¼Á´Ùà¤éҺ͡Çèҵͺ (pi^2)/4 ¹èФÃѺ §Ñé¹¼Á¶ÒÁÍÕ¡¢éÍä´éÁÑé¤ÃѺ --------------------- ÍÔ¹·Ôà¡Ãµ (x^2 - 2x + 4)^(-3/2) µÑé§áµè 1 ¶Ö§ 2 *** àËç¹·Õèà¤éÒá¹Ð¹Ó¤×Í ãËé x-1 = sqrt(3)*tan u **** ·Õè¨Ð¶ÒÁ¾Õè¡ç¤×Í ÁÕÇÔ¸Õ§èÒ¡ÇèÒ¹Õé ÁÑé¤ÃѺ à¾ÃÒÐÍÂÙè æ ¼Á¡ç¤§¤Ô´äÁèÍÍ¡ËÃÍ¡ÇèÒ µéͧãËéÍѹ¹Ùé¹à·èҡѺÍѹ¹Õé ã¤Ã¨Ð令Դ¶Ö§ ÅèФÃѺÇèÒµéͧãËéà·èҡѺ sqrt(3)* tan u Áѹà˹×ͨԹµ¹Ò¡ÒÃÁÒ¡àÅ ¤Ô´äÁèÍÍ¡¤ÃѺ ¢Íº¤Ø³¤ÃѺ¾Õè |
#6
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ÍéÍ! ¾Õè¡Ã¤ÃѺ ¢éÍ·Õè¼Á¶ÒÁäÇé¤ÃÑé§áá
¹èÐ ÁѹËÒáѹ¹Ð¤ÃѺ ·Õè¾Õè·ÓÃÙéÊÖ¡ÇèÒÁѹ ¨Ð¤Ù³¡Ñ¹ ----- ¢Íº¤Ø³¤ÃѺ |
#7
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Í×Á. ´Ù¼Ô´ä»¨ÃÔ§ æ ¤ÃѺ. ⨷ÂìÁѹËÒáѹ¹Õè¹Ð
à´ÕëÂÇÇèÒ§ æ ¨ÐÅͧ·Ó´ÙÍÕ¡·Õ áµè¾ÕèÇèÒÁѹ¡ç¤§äÁè·ÓàÊÃ稡ѹã¹ÍÖ´ã¨ËÃÍ¡ ¤§äÁèµèÓ¡ÇèÒ˹éÒ¹Ö§¹èÐÅèйР¡ÒÃà»ÅÕè¹µÑÇá»ÃµÃÕ⡳ÁѹÁÕ 3 ÃٻẺ ¤ÃѺ ¤×Í sin , sec , tan áµè¾Õè¨ÐºÍ¡á¤è 2 ÃٻẺ ¾Í à¾ÃÒжéÒ¹éͧ´Ùà¢éÒã¨ä´éÁѹ¡çäÁèµéͧ¨Ó ¤×ÍÅͧÁÑèÇ æ ´Ù áÅéÇà¢Õ¹ÊÒÁàËÅÕèÂÁÁØÁ©Ò¡ ÇèÒä´é¤èÒ´éÒ¹·ÕèàÃÒÁÕÍÂÙèÍÍ¡ÁÒ¨ÃÔ§ãËÁ. 1.ÃÙ» a^2 + b^2 x^2 ÍÂèÒ§¹ÕéàÃÒ¨ÐÊÁÁµÔãËé ax = b tan z 2.ÃÙ» a^2 - b^2 x^2 ÍÂèÒ§¹ÕéàÃÒ¨ÐÊÁÁµÔãËé ax = b sec z Êèǹ sin ¹Ñé¹Áѹ¨ÐÊÅѺ¡Ñº sec Åͧ¤Ô´´ÙÅСѹ µÑÇÍÂèÒ§. integrate(dx / 9 + 16x^2 ) ..........................# 1.ãËé 4x = 3 tan z ´Ñ§¹Ñé¹ tan z = 4x / 3 áÅéÇÇÒ´ÊÒÁà¡ÅÕèÂÁÁØÁ©Ò¡¢Öé¹ÁÒ ãÊè·Ø¡´éÒ¹ãËé¤Ãº 2. ¨Ò¡â¨·Âì ÊѧࡵÇèÒàÃÒ¨Ðáºè§¤èÒÍÍ¡à»ç¹ 2 ªØ´ ·Õè¨Ðà»ÅÕè¹ ¤×Í dx ¡Ñº 9 + 16x^2 3. ¨Ò¡¢éÍ 1. diff «Ð ¨Ðä´é dx = (3/4) (sec^2 z) dz ............(3.1) ¨Ò¡ÊÒÁàËÅÕèÂÁÁØÁ©Ò¡ã¹¢éÍ 2. àÃҨоºÇèÒ sqrt [ (9 + 16x^2 ) ] / 3 = sec z ´Ñ§¹Ñé¹ 9 + 16x^2 = 9 sec^2 z .............(3.2) 4. á·¹ (3.1) , (3.2) ŧ㹠# ¨Ðä´éÇèÒ integrate(dx / 9 + 16x^2 ) = (3/4)(sec^2 z dz) / 9 sec^2 z = (1/12) dz ...............(4) ´Ñ§¹Ñé¹àÃÒµéͧ·Ó¡Òà integrate (1/12) dz «Öè§ä´é (1/12) z + C áµè ¢éÍ 1. 4x = 3 tan z ´Ñ§¹Ñé¹ z = arctan(4x/3) «Öè§á·¹¤èÒŧ仨Ðä´é (1/12) arctan (4x/3) + C ¤ÃѺ. ÍÂèÒ§¢éÍ·Õè¹éͧÇèÒÁÒ ¡çá¤è¨Ñ´ÃÙ» à»ç¹ + sqrt(3)]^2+ (x-1)^2 ¨Ö§ÊÁÁµÔãËé x - 1 = [sqrt(3)] tan z 䧤ÃѺ. ¹éͧÍèҹ˹ѧÊ×Í ¤ÇÃàÅ×Í¡·ÕèÁѹà¢Õ¹§èÒÂ æ ¡è͹«Ô¤ÃѺ. ÁѹÁÕÍÂÙèàÅèÁ¹Ö§ ÍÍ¡ÁÒ¹Ò¹áÅéÇ ¾ÕèÍèÒ¹µÍ¹ÍÂÙè Á.»ÅÒÂÃÙéàÃ×èͧ·Õà´ÕÂÇ ¾ÕèàÅ diff + integrtate ¾Ç¡á¤Å·Ø¡ÃٺẺà»ç¹áµè Á. »ÅÒ ÁÒàÃÕ¹á¤Å I ·Õè ¨ØÌÒ¹Õèá·ºäÁèÍèÒ¹à¾ÔèÁàÅ à¢Õ¹ä´é§èÒ æ ÁÒ¡ æ àÅ ¤×Í à¹é¹·Óä´é¡è͹ ¨ÃÔ§ æ à¢ÒàÅÔ¡¾ÔÁ¾ì¢ÒÂ仹ҹáÅéÇ áµèËÅÒÂÇѹ¡è͹¾ÕèàËç¹µÒÁÃéҹ˹ѧÊ×Í·Õèä˹«Ñ¡áË觹ÕèÅèÐá¶Ç æ ¨ØÌÒ ¹ÕèÅèР˹éÒ»¡Áѹ¨Ðà»ç¹ÃÙ»¡ÃÒ¿ 3 ÁÔµÔ áººµÒ¢èÒ ¶éÒ¨ÓäÁè¼Ô´¹Ð »¡ÍÍ¡ÊÕàËÅ×ͧ»¹à¢ÕÂÇ áµè·ÕèàËç¹ÍѹãËÁèÃÙéÊÖ¡¨Ðà»ç¹ÍÍ¡ÊÕ¿éÒÍè͹¹Ð¤ÃѺ |
#8
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á¤Å»Õ˹Öè§àÃÕ¹à¡ÕèÂǡѺÍÐäúéÒ§¤ÃѺ
·Õè¾Õèà¤ÂºÍ¡ã¹¡ÃзÙé¡è͹ æ ¤×Í diff ¡Ñº integrate áÅéÇÁÕÍÂèÒ§Í×è¹ÍÕ¡ ÁÑé¤ÃѺ áÅéÇÁѹ¨Ð¶Ö§àÃ×èͧ͹ؾѹ¸ìÂèÍ àÇ¡àµÍÃì¡ÑºÍÔ¹·Ôà¡ÃµËÅÒªÑé¹ÁÑé¤ÃѺ ¶éÒà»ç¹ä»ä´éÍÂÒ¡ÃÙéÇèÒ»Õä˹ cal àÃÕ¹ ÍÐäà à¾ÃÒмÁä´éà¢éÒä»·Õè web ¢Í§ÇÔÈÇШØÌÒ ¼ÁäÁèà¨ÍÃËÑÊÇÔªÒ·Õèà»ç¹ cal àŤÃѺ -------- ·Õè¾ÕèºÍ¡ÁÒàÃ×èͧ sin sec tan ¼Á¾Í¨Ðà¢éÒ ã¨áÅéÇÅèФÃѺ àÁ×èÍ¡ÕéÅͧ价Ó⨷Âì¢éÍá¹Ç ¹Õé´Ù¡ç·Óä´éáÅéÇÅèФÃѺ ¢Íº¤Ø³ÁÒ¡¤ÃѺ ´Õã¨ÁÒ¡¤ÃѺ·ÕèÁÕ¤¹·Ó web ´Õ æ Ẻ¹Õé ªèÇ àËÅ×Íà´ç¡¸ÃÃÁ´Ò æ ÍÂèÒ§¼Áä´éàÊÁÍ |
#9
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Í×Á. Áѹ¡çÁÕËÅÒÂàÃ×èͧ¹Ð¤ÃѺ.
äÁèãªèá¤è diff ¡Ñº ÍÔ¹·Ôà¡ÃµÍÂèÒ§à´ÕÂÇËÃÍ¡ ÃÒÂÅÐàÍÕ´ÅÖ¡ æ¾Õè¨ÓäÁèä´é¤ÃѺ. áµè diff ¡Ñº ÍÔ¹·Ôà¡Ãµ·Ø¡ÃٻẺÁѹµéͧáÁ蹡è͹. ¹éͧàÍÒãËéáÁè¹à»ÃÕêÂÐä»àŤÃѺ. ¨Ðä´é©ÅØ à¾ÃÒÐà´ÕëÂǨÐä»àÃÕ¹àÃ×èͧÍ×è¹µèÍäÁèä´é á¹è¹Í¹ÇèÒàÃ×èͧ partial diff ËÃ×Í ÍÔ¹·Ôà¡ÃµËÅÒªÑé¹ ÁѹÁÕÍÂÙèá¹è æ à¾Õ§áµè¾Õè¨ÓäÁèä´éÇèÒÁѹÍÂÙèá¤Åä˹¤ÃѺ ¹éͧ仵éͧä»á¤ÃìÇèÒÍÐäÃàÃÕ¹»Õä˹ËÃÍ¡¤ÃѺ. á»êºà´ÕÂÇà´ÕëÂÇ¡çä´éàÃÕ¹ äÁèµéͧ¡ÅÑÇÅ×ÁËÃÍ¡ ÍèÒ¹ÁѹàÂÍÐ æ ÃѺÃÙéãËéÁÒ¡·ÕèÊØ´à·èÒ·Õè¹éͧ¨ÐÃѺÃÙéä´éÅèФÃѺ. à¾ÃÒе͹àÃÕ¹¨ÃÔ§ æ ÁѹàÃçÇ¡ÇèÒ ÁѸÂÁ 3 - 4 à·èÒàªÕÂǹФÃѺ. ÍèÒ¹¡Ñ¹á·ºäÁè·Ñ¹ ·Ñ駷ÕèàÇÅÒÇèÒ§¡çàÂÍÐ àÃÕ¹¡çá¤èà·ÍÁÅÐ 7 - 8 µÑÇàͧ áµè¨ÐãËéªÑÇÃì¡çä»´Ù˹ѧÊ×Íá¤Å¢Í§¤³ÐÇÔ·ÂÒÈÒʵÃì ·ÕèÈÙ¹Âì˹ѧÊ×ͨØÌÒ «Ô¤ÃѺ. ¾Ç¡á¤ÅµÍ¹¹Õé¾ÕèäÁè¤èÍÂä´éáµÐ¤ÃѺ. ¨Ðà¹é¹ä»·Ò§ discrete math ÍÂÙè à¾ÃÒÐÃÙéÊÖ¡ÇèÒÁѹ˹ء¡ÇèҾǡá¤ÅàÂÍÐàÅÂ. ÁÕàÃ×èͧÍèÒ¹áÅéǻǴËÑÇäÁèà¢éÒ㨧èÒ æ àÂÍÐàÅ äÁèàËÁ×͹á¤Å·ÕèÍèÒ¹àÁ×èÍäáçà¢éÒ㨷ѹ·ÕàÁ×è͹Ñé¹ |
#10
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discrete ¤×ÍÍÐäÃàËÃͤÃѺ áÅéÇÁѹʹءÂѧ䧤ÃѺ
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#11
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1. ãËé u = x áÅÐ dv = (sinx) / ( 1 + cos^x) dx
du = dx áÅÐ v = integrate( sin x / ( 1 + cos^x) d (cosx) / -sinx = -arctan( cos x ) 2. â´Â By part ¨Ðä´é ⨷Âì = -x arctanx + integrate( arctan (cosx) dx ) ******* 3. ·Õ¹ÕéÁÒ¾Ô¨ÒÃ³Ò function y = arctan x àÃÒÂѧ¤§¨Ó¡Ñ¹ä´éÇèÒ ¿Ñ§¡ìªÑ¹¤Ùè ¤×Í f(-x) = f(x) ¿Ñ§¡ìªÑ¹¤Õè ¤×Í f(-x) = - f(x) ¨Ð¾ºÇèÒ f(x) = acrtan x f(-x) = arctan(-x) = - arctan x ¨Ö§à»ç¹ ¿Ñ§¡ìªÑ¹ ¤Õè 4. ã¹á§è¢Í§¡ÒÃÍÔ¹·Ôà¡Ãµ àÃҨйÔÂÒÁÇèÒ f(x) à»ç¹¿Ñ§¡ìªÑ¹¤Ùè ¡çµèÍàÁ×èÍ integrate ¨Ó¡Ñ´à¢µµÑé§áµè -a ¶Ö§ a ¢Í§ f(x) dx ¨Ð = 2 integrate ¨Ó¡Ñ´à¢µµÑé§áµè 0 ¶Ö§ a ¢Í§ f(x) dx f(x) à»ç¹¿Ñ§¡ìªÑ¹¤Õè ¡çµèÍàÁ×èÍ integrate ¨Ó¡Ñ´à¢µµÑé§áµè -a ¶Ö§ a ¢Í§ f(x) dx ¨Ð = 0 5. ¨Ò¡â¨·Âì àÁ×èÍ àÃÒ ÍÔ¹·Ôà¡Ãµ¨Ò¡ x = 0 ¶Ö§ pi ¨Ðä´éÇèÒ cos x ÁÕ¤èÒ µÑé§áµè -1 ¶Ö§ 1 ¹Ñè¹àͧ ´Ñ§¹Ñé¹ integrate( arctan (cosx) dx ) ¨Ö§ÁÕ¤èÒ = 0 ¹Ñè¹àͧ 6. á·¹¤èÒ Å§ã¹¢éÍ 2. ¨Ö§ä´é = [ - ( pi)(-pi/4) ] - [ 0 ] = pi^2 / 4 ......Ans |
#12
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discrete math
¡ç ¤×Í ¤³ÔµÈÒʵÃìäÁèµèÍà¹×èͧ àªè¹ Number Theory, Combinatoric , Graph Theory à»ç¹µé¹. ʹءÂÑ§ä§ ËÃ×Í ¤ÃѺ. ÁѹÁÕ»ÑËÒãËé»Ç´ËÑÇàÂÍдդÃѺ. ¨ÐàÍÒ«Ñ¡ËÅÒ æ àÃ×èͧãËÁÅèФÃѺ ʹءÇèÒ á¤ÅËÃ×;ǡÍ×è¹ æ àÂÍÐàÅ |
#13
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áµè¼ÁÇèÒ á¤Å ʹءÊØ´áÅéǹРµÑé§áµèÍèÒ¹¢Í§ÁËÒÅÑ·ÕèËéͧÊÁØ´âçàÃÕ¹ÁÕãËéÍèÒ¹(¢³Ð¹ÕéÁ.4) á¤Å¹ÕèáËÅÐ ÃÙéàÃ×èͧÁÒ¡·ÕèÊØ´
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¡ÒáÅÒ¾ѹ¸Øì: àÁ×èÍàÍÒ»Õ 2542 à»ç¹»Õ°Ò¹ ¾ºÇèÒ ¢éÍÊͺ¤³Ôµ 1 »Ñ¨¨ØºÑ¹ ÂÒ¡ÃÒǡѺ ÊÁÒ¤Á¤³ÔµÈÒʵÃì »Õ 42 ¢éÍÊͺ¤³Ôµ 2 »Ñ¨¨ØºÑ¹ ÂÒ¡ÃÒǡѺ ¢éÍÊͺ¤³Ôµ 1 »Õ 42 ¢éÍÊͺÊÁÒ¤Á¤³ÔµÈÒʵÃì »Ñ¨¨ØºÑ¹ ÂÒ¡ÃÒǡѺ¢éÍÊͺâÍÅÔÁ»Ô¡ä·Â »Õ 42 ͹Ҥµ ¤³Ôµ 1 ¨Ð¡ÅÒÂà»ç¹âÍÅÔÁ»Ô¡ ¤³Ôµ 2 ¨Ð¡ÅÒÂà»ç¹ÊÁÒ¤ÁÏ áÅéÇ·Õ¹Õé ¢éÍÊͺâÍÅÔÁ»Ô¡ä·Â ¨Ð¡ÅÒÂà»ç¹ IMO ÁÑéÂÅèÐà¹Õè |
#14
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Int[x = 0 -> x = Pi, x sin x / (1+cos2x)] ᡪèǧà»ç¹ 0 -> Pi/2 áÅÐ Pi/2 -> Pi
¨Ò¡¤Ø³ÊÁºÑµÔ¢Í§ sin x áÅÐ cos x ÊÔ觷Õèµéͧ¡ÒèÐà·èҡѺ Int[x = 0 -> x = Pi/2, x sin x / (1+cos2x)] + Int[x = 0 -> x = Pi/2, (Pi - x) sin x / (1+cos2x)] = Int[x = 0 -> x = Pi/2, Pi sin x / (1+cos2x)] = Pi2/4 |
#15
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¹Ñè¹ä§ÁÒáÅéÇÇÔ¸Õ·Ó
ÁѹµéͧÍÂèÒ§¹Õé«Ô¤ÃѺ ¤èÍÂÊÁà»ç¹àÇçº Math ˹èÍÂ. ·º. äÁèÁÕ»ÑËÒ¤³ÔµÈÒʵÃìã´·ÕèäÁèä´éÃѺ¡ÒÃá¡é·ÕèºÍÃì´áË觹Õé (¶éÒà»ç¹»ÑËÒ·ÕèÁÕ¼Ùéá¡éÍÍ¡áÅéÇ) àÍÍ.ãªè ¾Õè¡çÍèÒ¹á¤Å»ÃÐÁÒ³. Á.4 ÁÑé§ ¡çà¾ÃÒÐÁѹ§èÒ¡ÇèÒàÃ×èͧÍ×è¹ æ ä§ ÍèÒ¹áÅéÇÃÙéàÃ×èͧ ÇèÒáµèŧÅ֡仨ÃÔ§ æ ¡çÂÒ¡¹Ð |
ËÑÇ¢éͤÅéÒ¤ÅÖ§¡Ñ¹ | ||||
ËÑÇ¢éÍ | ¼ÙéµÑé§ËÑÇ¢éÍ | Ëéͧ | ¤ÓµÍº | ¢éͤÇÒÁÅèÒÊØ´ |
⨷Âì Integral ¤èÐ ªèǤԴ·Õ¹Ðææææææææææææææææ | Ding Dong | Calculus and Analysis | 7 | 25 ¡Ã¡®Ò¤Á 2006 15:23 |
»ÑËÒªÔ§ÃÒ§ÇÑÅ¢éÍ·Õè 17: Definite Integral | warut | ¤³ÔµÈÒʵÃìÍØ´ÁÈÖ¡ÉÒ | 10 | 25 àÁÉÒ¹ 2006 19:59 |
complex integral ¤ÃѺ | Counter Striker | »ÑËÒ¤³ÔµÈÒʵÃì·ÑèÇä» | 2 | 19 àÁÉÒ¹ 2005 15:27 |
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