A m na game. Ƙidãyar m integrals

Daya daga cikin muhimman hakkokin sassan na ilmin lissafi bincike ne na game ilimin lissafi. Yana maida hankali ne akan mai fadi da filin abubuwa, inda na farko - shi ne m na game. Matsayin shi tsaye a matsayin wani key cewa shi ne har yanzu a makarantar sakandare ta bayyana karuwar al'amurra da dama, abin da ya bayyana hakan lissafi.

bayyanar

A duban farko, alama sarai na game zuwa zamani, Topical, amma a yi shi dai itace cewa da ya je da baya a 1800 BC. Home to hukumance dauke Misira kamar yadda bai isa mana a baya shaidar ta zama. Yana saboda rashin bayani, duk yayin da armashi, kawai a matsayin sabon abu. Ya sake tabbatar da mataki na kimiyya ci gaba da mutãnensu daga waɗanda sau. A karshe, cikin ayyukan da aka samu da tsoho Greek lissafi, dating daga 4th karni BC. Sun bayyana Hanyar amfani da inda m integral, jigon wanda ya samu girma ko yanki na wani curvilinear siffar (uku-girma da kuma biyu-girma jirgin sama, bi da bi). lissafi da aka bisa ga ka'ida da rabo daga cikin asali adadi a cikin infinitesimal aka gyara, bayar da cewa girma (yanki) aka riga aka sani zuwa gare su. A tsawon lokaci, da hanyar ya girma, Archimedes amfani da shi don samun fannin wani parabola. Similar lissafin a lokaci guda za su gudanar da atisayen a zamanin d China, inda suka kasance gaba daya m daga Greek yan'uwanmu kimiyya.

ci gaba

A gaba nasara a XI karni BC ta zama aiki na Arab masanin "wagon" Abu Ali al-Basri, da suka tura da iyakoki na riga aka sani, aka samu daga na game dabara domin kirga da asalin dũkiyõyinku da yawa da darajõji daga na farko zuwa na hudu, da ake ji wa wannan sani a gare mu shigar da hanya.
Zukatan yau ake ganin darajarsa ta d ¯ a Masarawa halitta mai ban mamaki Monuments ba tare da wani musamman kayan aikin, sai dai cewa na hannuwansu, amma an ba ikon mahaukaci masana kimiyya na lokaci babu kasa mai mu'ujiza? Idan aka kwatanta da na yanzu sau da rayukansu ze kusan m, amma da shawarar m integrals deduced a ko'ina da kuma amfani a yi domin kara bunkasa.

A mataki na gaba da ya faru a cikin XVI karni, a lokacin da Italian lissafi Cavalieri kawo basa hanya, wanda tsince Per Ferma. Wadannan biyu hali aza harsashin ga zamani na game ilimin lissafi, wanda aka sani a wannan lokacin. Su daura da Concepts bambantawa da kuma hadewa, wanda aka a baya gani kamar yadda kai dauke raka'a. By kuma manyan, da lissafi na cewa lokaci ya fragmented barbashi binciken zama da kansu, tare da iyaka amfani. Way to gama da kuma samun manufa daya ya kadai gaskiya a wannan lokacin, kuma gode masa, na zamani ilmin lissafi analysis samu damar girma da kuma ci gaba.

Tare da nassi daga lokaci ya canza kome da kome da na game alama ce da. By kuma manyan, shi ya sanya masana kimiyya suka a kansa hanya, misali, Newton amfani da wani square icon, wanda ya sa wani integrable aiki, ko kuma kawai sa tare. Wannan fidda dade har zuwa XVII karni, a lõkacin da wata alama ga dayan ka'idar ilmin lissafi analysis masanin kimiyya Gotfrid Leybnits gabatar da irin wannan hali saba mana. Elongated "S" ne a zahiri dangane da takardata wannan, Roman haruffa, tun suturta Naira Miliyan Xari primitives. The sunan na game samu godiya ga Jakob Bernoulli, bayan shekaru 15.

The m definition

A m na game dogara a kan definition daga cikin m, don haka mun yi la'akari da shi a da fari.

Antiderivative - ne kishiya aiki na wanda aka samu, a yi shi ne ake kira m. In ba haka ba: m aiki na d - shi ne mai aiki D, wanda shi ne wanda aka samu v <=> V '= v. Search m ne yin lissafi da m na game, da kuma aiwatar da kanta ne ake kira hadewa.

misali:

The aiki s (y) = y ^3, da kuma ta m S (y) = (y ^4/4).

A sa dukkan primitives na aiki - wannan shi ne wani m integral, denoted shi kamar haka: ∫v (x) DX.

By nagarta da cewa V (x) - ne kawai wasu m asali aiki, magana riqe: ∫v (x) DX = V (x) + C, inda C - m. A karkashin sabani akai tana nufin duk wani akai, tun da wanda aka samu shi ne sifili.

Properties

Da kaddarorin mallaki by m na game, da gaske dangane da definition da kaddarorin Kalam.
Ka yi la'akari da key maki:

• na game wanda aka samu daga cikin m ne m kanta da wani sabani akai C <=> ∫V '(x) DX = V (x) + C;
• wanda aka samu daga cikin na game da wani aiki ne na asali aiki <=> (∫v (x) DX) '= v (x).
• m aka dauka fita daga karkashin integral alamar <=> ∫kv (x) DX = k∫v (x) DX, inda k - shi ne mai sabani.
• integral, wanda aka dauka daga Naira Miliyan Xari da identically daidai da Naira Miliyan Xari integrals <=> ∫ (v (y) + w (y)) DY = ∫v (y) DY + ∫w (y) DY.

A karshe biyu Properties za a iya ƙarasa da cewa m integral ne mikakke. Saboda wannan, dole mu: ∫ (KV (y) DY + ∫ LW (y)) DY = k∫v (y) DY + l∫w (y) DY.

Don ganin misalai na kayyade mafita m integrals.

Dole ne ka samu na game ∫ (3sinx + 4cosx) DX:

• ∫ (3sinx + 4cosx) DX = ∫3sinxdx + ∫4cosxdx = 3∫sinxdx + 4∫cosxdx = 3 (-cosx) + 4sinx + C = 4sinx - 3cosx + C.

Daga cikin misali za mu iya cewa ba ka san yadda za su magance m integrals? Just sami duk primitives! Amma da search for ka'idojin tattauna a kasa.

Hanyar da Misalai

Domin warware integral, za ka iya koma ga wadannan hanyoyin:

• shirya ya dauki amfani da tebur.
• hadawa da sassan.
• hadedde ta maye gurbin da m;
• Summing up karkashin alamar bambanci.

alluna

A mafi sauki da kuma m hanya. A lokacin, ilmin lissafi analysis iya fariya quite m alluna, wanda rattaba kalma daga ainihin dabara na m integrals. A wasu kalmomin, akwai shaci samu har zuwa gare ka, kuma ba za ka iya kawai dauki amfani da su. A nan ne jerin babban tebur matsayi, wanda za a iya nuna kusan kowace misali, yana da wani bayani:

• ∫0dy = C, inda C - m.
• ∫dy = y + C, inda C - m.
• ∫y ⁿ DY = (y ^{n + 1)} / (n + 1) + C, inda C - akai, kuma n - lambar daban-daban daga hadin kai;
• ∫ (1 / y) DY = Ln | y | + C, inda C - m.
• ^{∫e y DY = e y + C} , inda C - m.
• ^{∫k y DY = (k y /} Ln k) + C, inda C - m.
• ∫cosydy = siny + C, inda C - m.
• ∫sinydy = -cosy + C, inda C - m.
• ∫dy / cos ² y = tgy + C, inda C - m.
• ∫dy / zunubi ² y = -ctgy + C, inda C - m.
• ∫dy / (1 + y ²⁾ = arctgy + C, inda C - m.
• ∫chydy = m + C, inda C - m.
• ∫shydy = chy + C, inda C - m.

Idan dole, yin kamar wata matakai kai integrand zuwa tabular ra'ayi da kuma ji dadin nasara. MISALI: ∫cos (5x -2) DX = 1 / 5∫cos (5x - 2) d (5x - 2) = 1/5 x zunubi (5x - 2) + C.

Bisa ga yanke shawara a fili yake cewa misali mai tebur integrand rasa multiplier 5. Mu ƙara da shi a layi daya tare da wannan halitta ta 1/5 zuwa janar magana bai canja ba.

Hadewa da sassa

La'akari da ayyukan biyu - z (y) da kuma x (y). Dole ne su zama ci gaba differentiable a kan ta yankin. A daya bambantawa Properties muna da: d (xz) = xdz + zdx. Hadawa garesu, muna samun: ∫d (xz) = ∫ (xdz + zdx) => zx = ∫zdx + ∫xdz.

Rewriting sakamakon lissafi, za mu samu da dabara, abin da ya bayyana Hanyar hadewa da sassa: ∫zdx = zx - ∫xdz.

Me ya sa yake dole? Gaskiyar cewa wasu daga cikin misalai shi ne mai yiwuwa a rage wuya, bari mu ce, don rage ∫zdx ∫xdz, idan karshen ne kusa da tabular form. Har ila yau, wannan dabara za a iya amfani da fiye da sau daya, domin mafi kyau duka sakamakon.

Yadda za a warware m integrals wannan hanya:

• dole yin lissafi ∫ (s + 1) e ^2s DS

∫ (x + ^{1) e 2s DS = {z =} s + 1, dz = DS, y = 1 ^{/ 2e 2s, DY = e 2x} DS} = ((s + 1) e ^2s) / 2-1 / 2 ∫e ^2s DX = ((s + 1) e ^2s) / 2-e ^2s / 4 + C;

• Dole ne lissafi ∫lnsds

∫lnsds = {z = lns, dz = DS / s, y = s, DY = DS} = slns - ∫s x DS / s = slns - ∫ds = slns -S + C = s (lns-1) + C.

Maye gurbin m

Wannan manufa na warware m integrals ba kasa a bukatar fiye da baya biyu, ko rikitarwa. A hanya ne kamar haka: Bari V (x) - da na game da wasu aiki v (x). A cikin taron cewa, a kanta na game a Example slozhnosochinenny zo, shi ne wata ila don samun rude da tafi da ba daidai ba hanya mafita. Don kauce wa wannan yi canji daga m x to z, a cikin abin da janar magana gani Sauki yayin da rike z dangane x.

A ilmin lissafi sharuddan, wannan shi ne kamar haka: ∫v (x) DX = ∫v (y (z)) y '(z) dz = V (z) = V (y -1 (x)), inda x = y ( z) - canzawa. Kuma, ba shakka, da kishiya aiki z = y ^-1 (x) cikakken bayyana dangantakar da dangantaka da canji. Muhimmanci rubutu - da bambanci DX dole maye gurbinsu da wani sabon bambanci dz, tun da canji na m a cikin m integral ya shafi sake mayar da shi a ko'ina, ba kawai a cikin integrand.

misali:

• dole ne samu ∫ (s + 1) / (s ² + 2s - 5) DS

Aiwatar da canzawa z = (s + 1) / (s ² + 2s-5). Sa'an nan dz = 2sds = 2 + 2 (s + 1) DS <=> (s + 1) DS = dz / 2. A sakamakon haka, da wadannan magana, wanda shi ne mai sauki yin lissafi:

∫ (s + 1) / (s ² + 2s-5) DS = ∫ (dz / 2) / z = 1 / 2ln | z | + C = 1 / 2ln | s ² + 2s-5 | + C;

• dole ne ka samu na game ∫2 ^s e ^S DX

Don shirya da sake rubutawa a cikin wadannan nau'i:

^{∫2 s e s DS = ∫ (} 2e) s DS.

Mun nuna a fakaice ta = 2e (sauyawa na shaida wannan mataki ne ba, shi ne har yanzu s), mu ba mu ga alama rikitarwa unshe, a asali tabular form:

^{∫ (2e) s DS = ∫a} s DS = wani s / lna + C = (2e) s / Ln (2e) + C = 2 ^s e ^s / Ln (2 + lne) + C = 2 ^s e ^s / (ln2 + 1) + C.

Summing up wani bambanci ãyã

By kuma manyan, wannan hanya na m integrals - tagwayen dan'uwan da manufa na canji na m, amma akwai bambance-bambance a kan aiwatar da rajista. Bari mu bincika a cikin mafi daki-daki.

Idan ∫v (x) DX = V (x) + C da kuma y = z (x), sa'an nan ∫v (y) DY = V (y) + C.

A lokaci guda dole ne mu manta da maras muhimmanci na game rikirkida, daga cikinsu:

• DX = d (x + wani), da kuma cikinsa - kowane m.
• DX = (1 / a) d (gatari + b), inda a - m sake, amma ba sifili.
• xdx = 1 / 2D (x ² + b);
• sinxdx = -D (cosx).
• cosxdx = d (sinx).

Idan muka yi la'akari da janar hali inda muka lissafta da m integral, misalai za a iya subsumed karkashin general dabara w '(x) DX = DW (x).

misalai:

• dole ne samu ∫ (2s + 3) ² DS, DS = 1 / 2D (2s + 3)

∫ (2s + 3) ² DS = 1 / 2∫ (2s + 3) ² d (2s + 3) = (1/2) x ((2s + 3) ²⁾ / 3 + C = (1/6) x (2s + 3) ² + C;

∫tgsds = ∫sins / cossds = ∫d (coss) / coss = -ln | coss | + C.

online taimako

A wasu lokuta, laifi na wanda za zama ko lalaci, ko wani gaggawa bukatar, za ka iya amfani da online tsokana, ko kuma wajen, don amfani mai kalkuleta m integrals. Duk da bayyana mawuyaci da rigima yanayin da integrals, da yanke shawara shi ne batun da takamaiman algorithm, wanda dogara ne a kan manufa da "idan ba ka aikata ba ... sa'an nan ...".

Hakika, musamman m misalai na irin wannan kalkuleta ba zai Master, kamar yadda akwai lokuta a wadda a yanke shawara yana samun wani tsari na wucin gadi "tilasta" ta gabatar da wasu abubuwa a cikin tsari, saboda sakamakon ne bayyananne hanyoyin da za a isa. Duk da rigima yanayin wannan sanarwa, shi ne gaskiya, kamar yadda lissafi, bisa manufa, wani m kimiyya, da kuma ta farko haƙiƙa ya ɗauki bukatar karfafawa da kan iyakoki. Lalle ne, ga mai santsi gudu-a theories ne da wuya sosai don motsawa sama da halittu farfadowa, don haka ba ɗauka cewa da misalai na warware m integrals, wanda ya ba mu - wannan shi ne tsawo da dama. Amma baya ga fasaha gefen abubuwa. A kalla duba lissafin, za ka iya amfani da sabis cikin abin da aka rubuta mana. Idan akwai bukatar for atomatik lissafi na hadaddun maganganu, sa'an nan ba su yi haquri, a mafi tsanani software. Ya kamata a kula da farko a kan yanayi MatLab.

aikace-aikace

Da hukuncin m integrals farko duba alama gaba daya ware daga gaskiya, domin shi ne da wuya a ga bayyane amfani da jirgin sama. Lalle ne, kai tsaye amfani da su a ko'ina za ka iya ba, amma su ne a dole matsakaici kashi a aiwatar da janyewar mafita amfani a yi. Saboda haka, hadewa na baya bambantawa, ta haka ne rayayye takara a aiwatar da warware lissafai.
Bi da bi, wadannan lissafai da tasiri kai tsaye a kan hukuncin inji matsaloli, yanayin lissafi da kuma thermal watsin - a takaice, duk abin da ya ƙunshi na yanzu da kuma siffata da nan gaba. M integral, misalai daga abin da muka riga muka bincika a sama, kawai maras muhimmanci a duban farko, a matsayin tushe da wani sashe mafi kuma mafi sabon binciken.