The kalekale yofunika. Masovedwe wa integrals kalekale

Mmodzi mwa zigawo zikuluzikulu kusanthula masamu ndi yofunika kakyulasi. Limafotokoza munda lonse la zinthu, pamene woyamba - ndi kalekale yofunika. Malo yaima monga omwe ali kusekondale limasonyeza ambiri ziyembekezo ndi mwayi limene limafotokoza masamu apamwamba.

maonekedwe

Koyamba, zikuoneka konse yofunika masiku ano, apakhungu, komabe likukhalira kuti iye anabwerera mu 1800 BC. Home kuti ankaona mwalamulo Iguputo siinafike ife umboni kale linakhalapo. Ndi chifukwa chosowa zambiri, yonseyi pabwino chabe monga chodabwitsa ndi. Iye kamodzinso zimatsimikizira mlingo wa chitukuko sayansi ya anthu a nthawi imeneyo. Pomaliza, ntchito anapezeka akale masamu Greek, chojambulidwa cha 4. m'ma BC. Amafotokoza njira ntchito kumene kalekale integral, akamanena za amene anali kupeza buku kapena dera mawonekedwe curvilinear (atatu azithunzi omwe tikunena ndi ziwiri ooneka enieni ndege, motero). mawerengedwe inachokera pa mfundo za gulu la chithunzi choyambirira m'zigawo kakang'ono, malinga buku (dera) kale Adawadziwitsa. M'kupita kwa nthawi, njira wakula, Archimedes ntchito kupeza dera parabola a. Similar kuwerengetsera nthawi yomweyo kuchita kokoka China wakale, kumene iwo anali palokha kwathunthu ku Greek sayansi anzake.

chitukuko

The yojambula lotsatira mu XI m'ma BC wakhala ntchito ya katswiri Arab "ngolo" Abu Ali al-Basri, amene anakankhira malire a kale, anali anachokera ku chilinganizo yofunika kwa iwo ankatha kuwerengetsera limanena za ndalama ndi digiri ku oyamba wachinayi, ntchito kwa izi kudziwika kwa ife kupatsidwa ulemu njira.
Maganizo a masiku ano amalemekezedwa ndi Aiguputo akale analenga zipilala chodabwitsa popanda zipangizo zapadera, kupatula kuti za manja awo, koma si mphamvu misala asayansi nthawi chimodzimodzi chozizwitsa? Poyerekeza ndi nthawi atsopano a moyo wawo ngati pafupifupi wosazindikira, koma kusankha integrals kalekale anadziŵa kulikonse ndi ntchito kuchita chitukuko zina.

Chotsatira zinachitika m'zaka XVI, pamene masamu Italy Cavalieri anabweretsa indivisible njira, amene anatola Per Ferma. Awiriwa umunthu maziko kwa yofunika kakyulasi ano, amene amadziwika pa mphindi. Iwo anamanga mfundo zosiyanasiyana komanso kusakanikirana, amene sanaonepo ngati mayunitsi kudzikonda zili. Mokulira wa masamu wa nthawi imeneyo anali zosamveka particles anapezazo alipo mwa iwo okha, ndi tingagwire. Njira kugwirizanitsa ndi kupeza mumafanana anali woona yekha pa nthawiyo, uyamiko kwa Iye, masiku kusanthula masamu ndi mwayi kukula ndi kukhala.

Ndi ndime ya nthawi zimasintha chirichonse ndi chizindikiro yofunika komanso. Mokulira zinaperekedwa asayansi amene mu njira yake Mwachitsanzo, Newton ntchito apakati mafano, amene anayika ndi ntchito integrable, kapena kungoti pamodzi. disparity uwu unakhalapo mpaka XVII atumwi, pamene chikhomo cha chiphunzitso lonse la masamu kusanthula wasayansi Gotfrid Leybnits anayambitsa khalidwe ozoloŵereka kwa ife. Elongated "S" kwenikweni zochokera kalata ya zilembo Roma, kuyambira amatanthauza Uwerenge primitives. Dzina la yofunika analandira kuyamika Jakob Bernoulli, patatha zaka 15.

The tanthauzo ofunda

The yofunika kalekale zimadalira tanthauzo la wosazindikira, kotero ife kuona malo oyamba.

Antiderivative - ndi osiyanitsidwa ntchito ya otumphukira, mu mchitidwe iwo akutchedwa achikale. Apo ayi: wosazindikira ntchito ya anthu d - ndi ntchito D, ndilo otumphukira ndime <=> V '= v. Search wosazindikira ndi kuwerengera kalekale integral, ndi njira yokha amatchedwa kusakanikirana.

Mwachitsanzo:

Ntchito m (Y) y n'zofanana ^3, ndipo S ake wosazindikira (Y) = (Y ^4/4).

Achikhalire a primitives onse kugwira ntchito - ichi ndi kalekale integral, amaimira izo motere: ∫v (x) dx.

Pamaziko a chakuti V (x) - ena okha choyambirira ntchito wosazindikira, akuti akugwirizira: ∫v (x) dx = V (x) + C, kumene C - zonse. Mu zonse umasinthasintha amatanthauza aliyense nthawi zonse, kuyambira otumphukira ake ndi ziro.

katundu

katundu wogwidwa ndi kalekale integral, makamaka zochokera tanthauzo ndi katundu wa ena ofanana.
Taganizirani mfundo zikuluzikulu:

• yofunika otumphukira wa wosazindikira ndi wosazindikira yokha kuphatikiza wopondereza zonse C <=> ∫V '(x) dx = V (x) + C;
• otumphukira ya tsinde la ntchito yake ndi ntchito choyambirira <=> (∫v (x) dx) '= vesi (x);
• zonse akutengedwa kuchokera pansi pa yofunika chizindikiro <=> ∫kv (x) dx = k∫v (x) dx, kumene K - ndi umasinthasintha;
• integral, wochokera Uwerenge identically wofanana Uwerenge integrals <=> ∫ (v (Y) + yakumadzulo (Y)) Dy = ∫v (Y) Dy + ∫w (Y) Dy.

The katundu ziwiri zapitazi Ndizachidziwikire kuti yofunika chikhalire ndi liniya. Chifukwa iyi, tili: ∫ (kv (Y) Dy + ∫ lw (Y)) Dy = k∫v (Y) Dy + l∫w (Y) Dy.

Kuona zitsanzo za kukonza njira integrals kalekale.

Muyenera kupeza ∫ yofunika (3sinx + 4cosx) dx:

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

Pa chitsanzo tinganene kuti simukudziwa mmene angathetsere integrals kalekale? Ingopeza primitives onse! Koma kufunafuna mfundo zina zomwe pansipa.

Njira ndi Zitsanzo

Pofuna kuthetsa integral, mungathe kugwiritsa ntchito njira zotsatirazi:

• chire kugwiritsira ntchito pa tebulo;
• kuphatikiza ndi mbali;
• Integrated ndi kuchotsa variable;
• Polongosola mwachidule pansi pa chizindikiro cha masiyanidwe a.

matebulo

Kwambiri ophweka ndiponso njira. Pa nthawi, kusanthula masamu akhoza kudzitama matebulo yaikulu kwambiri, zomwe sizinatchulidwe chilinganizo yaikulu ya integrals kalekale. M'mawu ena, pali zidindo anachokera kwa inu ndipo inu mukhoza kutenga mwayi. Nawu mndandanda wa malo waukulu tebulo, sipangakhalenso anasonyeza pafupifupi aliyense Mwachitsanzo, ali ndi njira:

• ∫0dy = C, kumene C - zonse;
• ∫dy y n'zofanana + C, kumene C - zonse;
• ∫y ^N Dy = (Y ^{N + 1)} / (n + 1) + C, kumene C - zonse, ndipo N - chiwerengero wosiyana mgwirizano
• ∫ (1 / Y) Dy = Ln | Y | + C, kumene C - zonse;
• ^{∫e Y Dy = mauthenga Y + C} , kumene C - zonse;
• ^{∫k Y Dy = (k Y /} Ln k) + C, kumene C - zonse;
• ∫cosydy = siny + C, kumene C - zonse;
• ∫sinydy = -cosy + C, kumene C - zonse;
• ∫dy / Ko ² Y = tgy + C, kumene C - zonse;
• ∫dy / tchimo ² Y = -ctgy + C, kumene C - zonse;
• ∫dy / (1 + Y ²⁾ = arctgy + C, kumene C - zonse;
• ∫chydy = manyazi + C, kumene C - zonse;
• ∫shydy = chy + C, kumene C - zonse.

Ngati ndi kotheka, kupanga angapo masitepe kutsogolera integrand maganizo ake tabular ndi kusangalala chigonjetso. CHITSANZO: ∫cos (5x -2) dx = 1 / 5∫cos (5x - 2) d (5x - 2) = 1/5 × tchimo (5x - 2) + C.

Malinga ndi chigamulo zikuonekeratu kuti Mwachitsanzo tebulo integrand sadziwa multiplier 5. Tiyenera kuwonjezera limodzi ndi akuchulukila izi 1/5 kuti mawu onse sanasinthe.

Kusakanikirana ndi Mbali

Taganizirani ntchito awiri - z (Y) ndi × (Y). Iwo ayenera kukhala mosalekeza differentiable pa ulamuliro wake. Mu umodzi katundu zosiyanasiyana tili: D (xz) = xdz + zdx. Kuphatikiza mbali zonse, ife kupeza: ∫d (xz) = ∫ (xdz + zdx) => zx = ∫zdx + ∫xdz.

Rewriting anthu aone kuti pali chifukwa ife titenga kachitidwe, amene amafotokoza njira kusakanikirana ndi mbali: ∫zdx = zx - ∫xdz.

Chifukwa chiyani kuli kofunikira? kuti ena mwa zitsanzo n'zotheka wosalira, tiyeni tinene, kuchepetsa ∫zdx ∫xdz, ngati otsiriza ali pafupi ndi mawonekedwe tabular. Komanso, chilinganizo ichi chikhonza kangapo zotsatira mulingo woyenera.

Mmene angathetsere kalekale integrals motere:

• koyenera kuwerengera ∫ (s + 1) e ^2s DS

∫ (× + ^{1) e 2s DS = {z =} s + 1, dz = DS, Y = 1 ^{/ 2e 2s, Dy = mauthenga 2X} DS} = ((s + 1) e ^2s) / 2-1 / 2 ∫e ^2s dx = ((s + 1) e ^2s) / 2-e ^2s / 4 + C;

• ayenera kuwerengetsa ∫lnsds

∫lnsds = {z = lns, dz = DS / s, Y = m, Dy = DS} = slns - ∫s × DS / s = slns - ∫ds = slns -s + C = m (lns-1) + C.

Kuchotsa variable

Mfundo imeneyi kuthetseratu integrals kalekale si zochepa mu ankafuna kuposa m'mbuyomu awiri, ngakhale zovuta. Njira motere: Tiyeni V (x) - ndi yofunika ena ndime ntchito (x). Kukachitika kuti palokha yofunika mu Chitsanzo slozhnosochinenny akubwera, zikuoneka kuti asokonezeka ndipo kupita cholakwika njira njira. Kupewa kusintha kumapeto kwa × variable mpaka Z, imene mawu ambiri zowoneka wosalira pokhalabe ndi z malingana x.

Kumbali masamu, izi motere: ∫v (x) dx = ∫v (Y (z)) Y '(z) dz = V (z) = V (Y -1 (x)), kumene × y n'zofanana ( z) - M'MALO. Ndipo, ndithudi, osiyanitsidwa ntchito z y n'zofanana ^-1 (x) kwathunthu chinafotokozanso ndi ubale wa zosintha. Zofunika cholemba - ndi masiyanidwe dx kwenikweni m'malo ndi latsopano masiyanidwe dz, kuyambira kusintha kwa variable mu yofunika kalekale kumaphatikizapo n'kubweretsa kulikonse, osati integrand lapansi.

Mwachitsanzo:

• ayenera kupeza ∫ (s + 1) / (s ² + 2s - 5) DS

Ikani m'malo z = (s + 1) / (s ² + 2s-5). Ndiye dz = 2sds = 2 + 2 (s + 1) DS <=> (s + 1) DS = dz / 2. Chifukwa, mawu akuti otsatirawa, omwe ndi zosavuta kuwerengera:

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

• muyenera kupeza yofunika ∫2 ^m CN ^s dx

Pofuna kuthana ndi kulilembanso mu mawonekedwe zotsatirazi:

^{∫2 m CN s DS = ∫ (} 2e) m DS.

Ife potanthauza ndi = 2e (m'malo ya mtsutso wa stepi ino si choncho, akadali S), timapereka athu ooneka ngati zovuta yofunika kuti zofunika tabular form:

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

Polongosola mwachidule ndi masiyanidwe chizindikiro

Mokulira njira integrals kalekale - m'bale amapasa mfundo za kusintha kwa variable, koma pali kusiyana mu ndondomeko ya kulembetsa. Tiyeni tione mwatsatanetsatane.

Ngati ∫v (x) dx = V (x) + C ndi Y = z (x), ndiye ∫v (Y) Dy = V (Y) + C.

Pa nthawi yomweyo tiyenera kukumbukira zazing'ono masinthidwe integral, Pakati:

• dx = D (× + a), ndipo m'mene - aliyense nthawi zonse;
• dx = (1 / munthu) d (nkhwangwa + b), kumene - zonse, koma osati ziro;
• xdx = 1 / 2D (× ² + b);
• sinxdx = -d (cosx);
• cosxdx = D (sinx).

Ngati ife taganizirani ambiri komwe ife kuwerengera kalekale integral, zitsanzo akhoza subsumed pansi chilinganizo ambiri yakumadzulo '(x) dx = dw (x).

zitsanzo:

• ayenera kupeza ∫ (2s + 3) ² DS, DS = 1 / 2D (2s + 3)

∫ (2s + 3) ² DS = 1 / 2∫ (2s + 3) ² D (2s + 3) = (1/2) × ((2s + 3) ²⁾ / 3 + C = (1/6) × (2s + 3) ² + C;

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

Online thandizo

Nthawi zina, vuto la umene ukhoza kapena ulesi, kapena Akufunika, mungathe kugwiritsa ntchito chimakulimbikitsani online, kapena kani, ntchito Chiwerengero integrals kalekale. Ngakhale zovuta imaonekera ndipo maganizo chikhalidwe cha integrals pa chisankho umamumvera aligorivimu awo enieni, ozikidwa pa mfundo ya "ngati inu simutero ... ndiye ...".

Kumene, zitsanzo makamaka zosamvetsetseka umenewu Chiwerengero sadzakhala kuzidziwa bwino, ngati pali milandu imene chisankho ali kupeza chongopeka "amakakamizidwa" mwa kuyambitsa zinthu zina pochita chifukwa zotsatira zake zimakhala zoonekeratu njira kuwafika. Ngakhale chikhalidwe maganizo a mawu awa, izo nzoona, ngati masamu, mu mfundo ndi sayansi umboni, ndipo cholinga chake chachikulu amaona kufunika kwa mphamvu malire. Inde, chifukwa yosalala amathamanga mu ziphunzitso ndi zovuta kusunthira ndi kusintha, choncho musaganize kuti zitsanzo za kuthetsa integrals kalekale, amene anatipatsa ife - ichi ndi kutalika mwayi. Koma ku mbali ya luso zinthu. Osachepera kukayendera anapezazo, mukhoza kugwiritsa ntchito utumiki umene unalembedwa kwa ife. Ngati pali kufunika mawerengedwe basi mawu zovuta, ndiye iwo alibe amachita mapulogalamu lalikulu. Kodi kulabadira makamaka chilengedwe MatLab.

ntchito

Zimene integrals kalekale koyamba zikuwoneka kwathunthu wopatulika kuchokera ku chenicheni, chifukwa n'kovuta kuona ntchito zoonekeratu ndege. Ndithudi, mwachindunji kugwiritsa ntchito iwo kulikonse inu sangathe, koma ndi yofunika wapakatikati amafotokozera mu ndondomeko ya achire njira ntchito mchitidwe. Motero, kuphatikiza zosiyanasiyana kumbuyo, motero mwachangu nawo ndondomeko za kuthetsa maikwezhoni.
Nayenso maikwezhoni awa ndi zimakhudza mwachindunji pa chisankho cha mavuto makina, njira yodutsamo mawerengedwe ndi matenthedwe madutsidwe - Mwachidule, chirichonse chimene chimapanga panopa ndi kutchukitsa m'tsogolo. Yosatha integral, zitsanzo zimene takambirana pamwambapa, zazing'ono yekha pa Koyamba, monga m'munsi kuti achite zambiri kutulukira zinthu zatsopano.