PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA I. Aìristo Olokl rwma 2. Orismèno Olokl rwma 3. Diaforetik èkfrash tou aìristou oloklhr matoc H Sunˆrthsh F () = 4. Olokl rwma AntÐstrofhc Sunˆrthshc f(t) dt Anaplhrwt c Kajhght c : Dr. Pappˆc G. Alèandroc
. Aìristo Olokl rwma (indefinite integral) Orismìc: 'Estw f mia sunˆrthsh orismènh se èna diˆsthma. Arqik sunˆrthsh parˆgousa thc f sto onomˆzetai kˆje sunˆrthsh F pou eðnai paragwgðsimh sto kai isqôei: F () = f(), gia kˆje. H ènnoia thc arqik c sunˆrthshc eðnai mða ènnoia pou orðzetai se diˆsthma kai ìqi se ènwsh diasthmˆtwn. ApodeiknÔetai ìti kˆje suneq c sunˆrthsh se èna diˆsthma èqei arqik sunˆrthsh sto diˆsthma. H arqik sunˆrthsh miac sunˆrthshc f (an upˆrqei), den eðnai monadik. Orismìc: To sônolo ìlwn twn paragous n miac sunˆrthshc f s> èna diˆsthma onomˆzetai aìristo olokl rwma thc f sto. Dhlad h F () + c eðnai to aìristo olokl rwma thc f(). f() d = F() + c, c R. : to sômbolo thc olokl rwshc, f : h oloklhrwtèa sunˆrthsh, : h metablht olokl rwshc, d : diaforikì tou. To sômbolo d dhl nei ìti h metablht olokl rwshc eðnai. Genikˆ me df() sumbolðzoume to diaforikì mðac sunˆrthshc f kai eðnai df() = f ()d. c : h stajerˆ olokl rwshc. Ac shmei soume ed ìti: (a) f () d = f() + c, c R ( ) (a) Sumbatikˆ: f() d = f(). Sunèpeia tou orismoô tou aìristou oloklhr matoc kai twn kanìnwn parag gishc eðnai oi e c dôo idiìthtec : λ f() d = λ f() d, λ R, [f() ± g()] d = f() d ± g() d kai genikˆ [λ f() ± µ g()] d = λ f() d ± µ g() d, λ, µ R. H diadikasða eôreshc tou aìristou oloklhr matoc eðnai h antðstrofh thc diadikasðac thc parag gishc. Ta aìrista oloklhr mata merik n basik n sunart sewn dðnontai ston parakˆtw pðnaka: 2
PÐnakac Basik n Aìristwn Oloklhrwmˆtwn. 2. 3. 4. 5. 6. 7. 8. 9. 0.. 2. 3. 4. 5. 6. d = + + + c,, > 0 d = ln + c, 0 a d = ln a a + c, 0 < a e d = e + c sin d = cos + c cos d = sin + c (kπ cos 2 d = tan + c, π 2, kπ + π ), k Z 2 sin 2 d = cot + c, (kπ, (k + ) π), k Z { a2 d = arcsin ( ) a + c, 2 arccos ( ) a + c2, { a 2 + 2 d = a arctan ( ) a + c, a arccot ( ) a + c2, a > 0, ( a, a) a 0 ( a2 + d = ln + ) a 2 + 2 + c, a 0 2 2 a 2 d = ln + 2 a 2 + c, > a > 0 sinh d = cosh + c cosh d = sinh + c cosh 2 d = tanh + c sinh 2 d = coth + c, 0 3
Idiìthtec tou Aìristou Oloklhr matoc - Kanìnec Olokl rwshc. d() = d( + ), d() = d( ), d() = d(), d() = ( d. ) 2. Sta oloklhr mata dunˆmewn prèpei to diaforikì na eðnai ìmoio me bˆsh thc dônamhc. ( + 5) 3 d = ( + 5) 3 ( + 5)4 d( + 5) = + c. 4 3. Sta oloklhr mata trigwnometrik n sunart sewn prèpei to diaforikì na eðnai ìmoio me to tìo. συν( + ) d = συν( + ) d( + ) = ηµ( + ) + c. 4. Sta oloklhr mata dunˆmewn tou e prèpei to diaforikì na eðnai ìmoio me ton ekjèth. e 2 d = e 2 d(2) = 2 2 e2 + c. Oloklhr mata thc morf c f () f() d kai f() f () d f () f() d = ln f() + c. f() f () d = f() df() = f 2 () 2 + c. 4
2. Orismèno Olokl rwma (definite integral) Olokl rwsh katˆ Riemann To ìrio tou ajroðsmatoc S n upˆrqei sto R kai eðnai aneˆrthto apì thn eklog twn endiˆmeswn shmeðwn ξ k. To ìrio autì onomˆzetai orismèno olokl rwma thc suneqoôc sunˆrthshc f. Dhlad : β n f() d = lim S n = lim f(ξ k ) k. n k 0 k= Je rhma thc Mèshc Tim c tou OloklhrwtikoÔ LogismoÔ An h sunˆrthsh f() eðnai suneq c sto [, β], upˆrqei èna toulˆqiston shmeðo ξ ston (, β), tètoio ste β f() d = f(ξ)(β ). To Jemeli dec Je rhma tou OloklhrwtikoÔ LogismoÔ An h sunˆrthsh f() eðnai suneq c sto [, β] kai F () eðnai mia parˆgousa thc f() [F () = f()], tìte β f() d = [F()] β = F(β) F(). - To je rhma autì ja qrhsimopoieðtai gia ton upologismì twn orismènwn oloklhrwmˆtwn. - To [, β] kaleðtai diˆsthma olokl rwshc me kˆtw ìrio kai pˆnw ìrio β. - Oi arijmoð kai β onomˆzontai ìria olokl rwshc. - H sunˆrthsh f lègetai oloklhr simh sto diˆsthma [, β]. 'Opwc prokôptei apì ton orismì tou oloklhr matoc, kˆje sunˆrthsh f suneq c sto [, β] eðnai oloklhr simh sto [, β]. - To orismèno olokl rwma eartˆtai apì th sunˆrthsh f kai ta ˆkra kai β kai ìqi apì thn << onomasða >> thc metablht c olokl rwshc. To grˆmma eðnai mða metablht (h metablht olokl rwshc) pou mporeð na antikatastajeð apì opoiod pote ˆllo grˆmma. 'Etsi èqoume: To orismèno olokl rwma β f() d = β β f() d f(t) dt. eðnai pragmatikìc arijmìc opìte ( β f() d ) = 0, se antðjesh me to aìristo olokl rwma f() d pou eðnai èna sônolo sunart sewn. 5
Idiìthtec tou orismènou oloklhr matoc. β [f() ± g()] d = β f() d ± β g() d kai genikˆ β [λ f() ± µ g()] d = λ β f() d ± µ β g() d, λ, µ R. 2. β c f() d = c β f() d 3. β f() d = γ f()d + β f() d γ 4. β f() d = f() d β 5. f() d = 0 6. β f() d β f() d 7. f() 0, [, β] β f() d 0 8. f() g(), [, β] β f() d β g() d. 'Estw mia sunˆrthsh f suneq c sto [, β] kai Ω to qwrðo pou perikleðetai apì thn C f, ton ˆona kai tic eujeðec = kai = β. Tìte: An f() 0 sto [, β], ja eðnai E(Ω) = β f() d. An f() 0 sto [, β], ja eðnai E(Ω) = β f() d. An f den diathreð stajerì prìshmo sto [, β], tìte E(Ω) = β f() d. 'Estw oi suneqeðc sunart seic sto [, β] kai Ω to qwrðo pou perikleðetai apì tic grafikèc parastˆseic twn f kai g kai tic eujeðec = kai = β. Tìte: An f() g() sto [, β], ja eðnai E(Ω) = β [f() g()] d. An f() g() sto [, β], ja eðnai E(Ω) = β [g() f()] d. An h diaforˆ f() g() den diathreð stajerì prìshmo sto [, β], tìte E(Ω) = β f() g() d. Prìtash: An h ˆrtia sunˆrthsh f eðnai oloklhr simh sto [, ], tìte f() d = 2 0 f() d. Prìtash: An h peritt sunˆrthsh f eðnai oloklhr simh sto [, ], tìte f() d = 0. 6
3. Diaforetik èkfrash tou aìristou oloklhr matoc H Sunˆrthsh F() = f(t) dt An sto orismèno olokl rwma jewr soume to èna ˆkro eleôjero wc proc thn metablht, èqoume To olokl rwma mporeð na pˆrei th morf f() d. f() d = F () F (). An t ra jèsoume c = F (), tìte to parapˆnw olokl rwma gðnetai akrib c to aìristo olokl rwma: f() d = F () + c = f() d. Je rhma: An f eðnai mia suneq c sunˆrthsh se èna diˆsthma kai shmeðo tou diast matoc, tìte h sunˆrthsh F () = eðnai mia parˆgousa thc f sto. Dhlad isqôei: f(t) dt,, ìpou a eðnai stajerˆ kai mia metablht f(t) dt = f(),. Me ˆlla lìgia, o rujmìc aôhshc tou embadoô F () eðnai Ðswc me thn tim thc f sto. Parathr seic: Apì to parapˆnw je rhma prokôptei ìti: i. AnagkaÐa proüpìjesh gia thn Ôparh thc F eðnai h sunèqeia thc f se diˆsthma. H F orðzetai se diˆsthma kai ìqi se ènwsh diasthmˆtwn. ii. Apì ton orismì thc F kˆje tim thc eðnai orismèno olokl rwma. F ( 0 ) = 0 f(t) dt 7
iii. Gia ton orismì thc F mporoôme na epilèoume opoiad pote - kˆtw ˆkro oloklhr matoc arkeð na eðnai stajerì stoiqeðo tou. Diaforetik epilog tou mac dðnei diaforetik arqik sunˆrthsh F thc f. 'Omwc ìlec oi sunart seic afoô eðnai arqikèc thc f, diafèroun katˆ mia stajerˆ c opoða mporeð na upologisteð: 'Estw 2 kai, 2 kai oi sunart seic: Tìte isqôei: F () F 2 () = F () = f(t) dt, F 2 () = f(t) dt f(t) dt = f(t) dt + 2 2 f(t) dt. 2 f(t) dt = iv. H F eðnai paragwgðsimh kai suneq c sto, epðshc h F eðnai suneq c sto. v. vi. f(t) dt = f(t) dt = f(t) dt = f(). f(t) dt = f(t) dt + f(). 2 f(t) dt = c,. vii. An sth jèsh tou, upˆrqei ˆllh sunˆrthsh F () = g() f(t) dt, tìte h parag gish gðnetai wc e c: g() F () = f(t) dt = f [g()] g () ìpou g() sunˆrthsh paragwgðsimh. An h sunˆrthsh g() eðnai kˆtw ˆkro tou oloklhr matoc, prin paragwgðsoume th fèrnoume sto ˆnw ˆkro allˆzontac to prìshmì tou oloklhr matoc. viii. An h sunˆrthsh f eðnai orismènh se ènwsh ènwn diasthmˆtwn, ta kai, prèpei na an koun sto Ðdio diˆsthma. Sunep c, an mac zhthjeð to pedðo orismoô miac sunˆrthshc F () = g() f(t) dt, brðskoume to sônolo sto opoðo h f(t) eðnai suneq c kai apaitoôme h parˆstash g() na an kei se autì. An h f(t) orðzetai se ènwsh diasthmˆtwn, apaitoôme h g() na an kei sto Ðdio diˆsthma me autì sto opoðo an kei to. i. An h sunˆrthsh F () orðzetai san: F () = g() f(t) dt h() tìte h parˆgwgìc thc upologðzetai afoô spˆsoume to olokl rwma se ˆjroisma dôo ˆllwn me koinì ˆkro opoiod pote arijmì pou na an kei sto diˆsthma sto opoðo h f eðnai suneq c, dhlad g() F () = f(t) dt + f(t) dt dhlad isqôei ìti: h() F () = f [h()] h () + f [g()] g (). 8
Parˆdeigma. Na breðte to pedðo orismoô kai thn parˆgwgo thc sunˆrthshc F () = 0 t t 2 dt. LÔsh H sunˆrthsh f(t) = t eðnai suneq c sto sônolo t 2 D f = (, ) (, ) (, + ). Gia na orðzetai h F prèpei na arkeð ta ˆkra olokl rwshc 0, na an koun sto Ðdio diˆsthma tou pedðou orismoô thc f. Epeid to 0 (, ) prèpei to (, ). 'Ara D F = (, ). Gia kˆje (, ) èqoume F () = 0 t t 2 dt = 2. Parˆdeigma 2. Na breðte to pedðo orismoô kai thn parˆgwgo thc sunˆrthshc F () = 2 2 LÔsh e t t dt. H sunˆrthsh f(t) = et t eðnai suneq c sto sônolo D f = (, 0) (0, + ). To pedðo orismoô thc g() = 2 eðnai D g = R kai thc h() = eðnai 2 D h = (, 2) (2, + ). 'Ara, an kei sto pedðo orismoô thc F an kai mìno an (a) D g D h (, 2) (2, + ) kai (a) oi g() kai h() an koun sto pedðo orismoô thc f. Epomènwc: 2 2 < 0 2 < 0 2 2 > 0 2 > 0 'Ara, D F = (0, ) (2, + ). IsqÔei ìti: 2 0 < < < 2 2 < 0 > > 2 (0 < < > 2). F () = f [h()] h () + f [g()] g (), ìpou f(t) = et t, g() = 2, h() = 2. F () = 2 2 = e 2 2 = e 2 e t t dt ( = e 2 2 ( 2) 2 2 2 + e 2 (2 ). ( ) + e2 2 2 ( 2 ) = ) ( 2) + e2 2 (2 ) = 9
4. Olokl rwma AntÐstrofhc Sunˆrthshc Gia to olokl rwma λ κ f () d - jètoume f () = u = f(u), d = f (u) du. Ta nèa ìria olokl rwshc eðnai: - gia = κ f(u) = κ u = f (κ) = kai - gia = λ f(u) = λ u = f (λ) = β. Opìte λ f (λ) f () d = u f (u) du = κ f (κ) Sto diˆsthma [, β] h f prèpei na eðnai paragwgðsimh kai << - >>. β β u f (u) du = f () d. QwrÐo pou orðzetai apì thn grafik parˆstash thc antðstrofhc sunˆrthshc f, ton ˆona kai tic eujeðec = kai = β. An f () 0 tìte: E(Ω) = λ f () d κ λ β E(Ω) = f () d = f () d. κ 0