Ke indefinite integral. Computation o indefinite integrals

Kekahi o na kumu pauku o ka makemakika Ka Ikepili o ka integral calculus. Ua uhi ka laulā kahua o ka mea, ma ka mea mua - ka mea, o ka indefinite integral. Position ia ihola ke kumu me ke ki i mea no ma ke kula kiʻekiʻe hōʻike he ulu ka helu ana o ka manaʻolana 'ia a me ka pōmaikaʻi, a wehewehe kiekie makemakika.

helehelena

Ma mua kilohi, ka mea pono loa integral i ke kālā no, topical, akā, ma ka hoʻomaʻamaʻa mea, huli mai i ka mea, i hele mai hoʻi i loko o 1800 BC. i ka Home kauoha aku manaoia Aigupita like i ole hiki iā mākou hoike mua o kona ola. Ia ma muli o ka nele o ka 'ike, i ka manawa a pau hoʻonoho wale me he kŰia. He kekahi manawa hou hōʻoia ana i ka nui o ka 'epekema? Acaeoey o na kanaka a pau o ia mau manawa. Eia hoi, i loaʻa nā hana i ke kahiko Greek mathematicians, ka launa pū, mai ka 4th kenekulia BC. Ka mea, e wehewehe i ka hana hoʻohana ma o ka indefinite integral, i ka i loko o ka a ua, e huli i ka buke a me ka wahi o ka curvilinear shape (ekolu-dimensional a me nā-dimensional pelane, niioaaonoaaiii). i ka ho omaulia i ka nānā 'ana ma luna o ka loina hooponopono o ka papa o ka palapala huahelu i loko o infinitesimal eiiiiiaiou, aka i ka buke (wahi) ua, ua ike ia ia. Over manawa, i ka iaoiaeii ua ulu, Archimedes hoʻohana mea, e imi i ka wahi o ka parabola. Similar hoʻomaulia i ka ia manawa e hoʻomaka kino ma Makamua China, i kahi o ka mea, ua loa kuokoa, mai ka olelo Helene, o ka hoa 'epekema.

ulu ana

I kekahi Hopu i loko o ka XI kenekulia BC ua lilo i ka hana ana o ka mea ¶ʻO'Araba haumāna "hale" Abu Ali AL-Basri, ka poe i kipaku i nā palena o ka uaʻike, i loko, mai ka integral haʻilula no ka helu ana i ka puu o na dala a me degere, mai ka mua a hiki i ka ha, ana no keia ike ia mai hoʻokomo hana.
Manaʻo o kēia lā e mahalo ia e na ko Aigupita kahiko i hana mai i na mea kupaianaha kia hoʻomanaʻo i ole kekahi hana i na mea paahana, koe wale no ka mea o ko lākou mau lima, akā, ua mea i ka mana hehenaʻepekema o ka manawa, aole emi i ka hana mana? Ke hoʻohālikelike me ka papa manawa o ko lākou mau ola he kokoke primitive, akā, no ka olelo hooholo o ka indefinite integrals ho oholo wahi, a hoʻohana 'ia i loko o ka hana pono no ka hou' ana.

I kekahiʻanuʻu i wahi i loko o ka XVI kenekulia, i ka wa a ka Italia mathematician Cavalieri lawe indivisible iaoia, a lalau aku la ia Per Ferma. Kēia mau kanaka i kau aku i ke kahua no ka mea ke kālā no integral calculus, a uaʻikeʻia ma ka minute. A nakinaki lākou i ka manaÿo o ka differentiation a me ka hoʻohui ', i i mua ike i ka hoʻoponopono-na huahelu. Ma a me ka nui, me ka makemakika o ia manawa ua mamala (particles) hoʻoholo 'nei ma o lakou iho, me ka i kaupalena' ana. Way, e hui me ka loaʻa, he pono ole ka lepo i ka oiaio wale i ka manawa, i ke aloha ia ia, i ka laekahi makemakika Ka Ikepili i ka manawa e ulu, a ulu.

Me ka hooholo ia o ka manawa hoʻololi na mea a pau, a me ka integral hōʻailona like hoʻi. Ma a me ka nui, ka mea i hoʻohikiʻia ka noiʻi i loko o kona mau aoao, no kekahi laʻana, Newton hoʻohana i ka huinahalike ikona, a kau i ka integrable hana, a me wale i kahi hookahi. Keia disparity ana a hiki i ka XVII kenekulia, ka wā i ka hiʻona'āina no ka mea a pau kumumanaʻo o ka makemakika Ka Ikepili nä känaka 'epekema Gotfrid Leybnits hoʻolauna ia ke ano kama'āina iā mākou. Elongated "S" ma muli o kēia maoli palapala he Roma Hawaiʻi, mai Kû ka huina o primitives. Ka inoa o ka integral loaa i ke aloha a hiki i Jakob Bernoulli, ma hope o 15 makahiki.

Ka olelo i ka ho'ākāka '

Ke indefinite integral hilinaʻi nui ma luna o ka wehewehena o ka primitive, no laila mākou e noonoo ia i loko o ka wahi mua.

Antiderivative - o ka inverse hana o ka derivative, ma ka hoʻomaʻamaʻa ka mea, ua kapaia primitive. I ole ia: primitive hana o ka b - mea he papa D, i mea ka derivative V <=> V '= v. Search primitive mea e hoʻomaulia i ka indefinite integral, a me ke kaʻina iho ua kapaia hoʻohui '.

Eia kekahi laʻana:

Ke kuleana pili i ke (y me) = y me ^3, a me kona primitive S (y me) = (y me ^4/4).

Ka 'ia paha o nā primitives o ke kuleana pili i - keia mea he indefinite integral, denoted mea like penei: ∫v (m) dx.

Aie i ka mea i V (m) - Eia kekahi o na palapala primitive mana, aia no he hopuna'ōlelo: ∫v (m) dx = V (m) + C, kahi C - ikaika. Ma lalo o ka ākeʻakeʻa kumu a eia i pili i kekahi ikaika, mai kona derivative mea Aʻohe.

waiwai

Nā waiwai i uluhia e ka indefinite integral,ʻano nui ma muli o ka mana a me ka waiwai o nā mea loaʻa.
E hoomanao i ka mea ki kumu:

• integral derivative o ka primitive mea primitive ia iho, eia ka ākeʻakeʻa kumu ikaika C <=> ∫V '(m) dx = V (m) + C;
• derivative o ka integral o ka papa o ka palapala kuleana pili i <=> (∫v (m) dx) '= V (m);
• ikaika ua lawe mai, mai lalo o ka integral hōʻailona <=> ∫kv (m) dx = k∫v (m) dx, kahi K - Ua ākeʻakeʻa kumu;
• integral, a ua laweʻia mai i ka huina o ka identically like me ka huina o nā integrals <=> ∫ (V (y me) + w, (y me)) dy = ∫v (y me) dy + ∫w (y me) dy.

Ka hope mau waiwai hiki e manao ia ka indefinite integral ka laina. Manawa a hiki i keia, ua i: ∫ (kv (y me) dy + ∫ lw (y me)) dy = k∫v (y me) dy + l∫w (y me) dy.

E ike examples o hoʻopunipuni pāʻoihana indefinite integrals.

Pono e loaʻa ka integral ∫ (3sinx + 4cosx) dx:

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

Mai ka hana, ua hiki ke manao nei makou, aole oe i ike pehea e hoʻoponopono i indefinite integrals? Iki ka huli a pau na primitives! Akā, i ka huli 'ana no nā kumu kūkā mai ma lalo nei.

Nā Papahana a me nā ano he kumu hoʻohālike

I mea e hoʻoponopono i ka integral, e hiki, Kamuela i ka kēia kiʻina hana:

• makaukau e lawe ka lanakila o ka papaʻaina;
• ka hoʻononiakahiʻana ma ka māhele;
• noniakahi ma ke kūapoʻana i ka ee iaaanu aey;
• summing mai ma lalo o ka hoailona o ka palena pau.

papa

I ka loa a me ka poe noonoo ole ka puaʻa ala. Ma ka manawa, makemakika Ka Ikepili ke kaena loa ai nui papa, i ka pela mai ka walaʻauʻana haʻilula o indefinite integrals. Ma na olelo e ae, he mau UAAIIeIIeX loko mai iā 'oe, a' oe hiki wale lawe ka lanakila o ia. Eia ka inoa o ka papaʻaina nā kūlana hana, i hiki ke hōʻikeʻaneʻane kela manawa, i ka pāʻoihana.

• ∫0dy = C, kahi C - ikaika;
• ∫dy = y me + C, kahi C - ikaika;
• ∫y ^N dy = (y me ^{N + 1)} / (N + 1) + C, kahi C - he mau, a me N - helu okoa, mai ke kuikahi;
• ∫ (1 / y me) dy = Ln | y me | + C, kahi C - ikaika;
• ^{∫e y me dy = E y me + C} , kahi C - ikaika;
• ^{∫k y me dy = (k y me /} Ln k) + C, kahi C - ikaika;
• ∫cosydy = siny + C, kahi C - ikaika;
• ∫sinydy = -cosy + C, kahi C - ikaika;
• ∫dy / cos ² y me = tgy + C, kahi C - ikaika;
• ∫dy / hewa ² y me = -ctgy + C, kahi C - ikaika;
• ∫dy / (1 + y me ²⁾ = arctgy + C, kahi C - ikaika;
• ∫chydy = hilahila + C, kahi C - ikaika;
• ∫shydy = chy + C, kahi C - ikaika.

Inā pono, e ka mau o nāʻanuʻu alakai integrand i ka tabular nānaina, a nanea i ka lanakila. Eia kekahi laʻana: ∫cos (5x -2) dx = 1 / 5∫cos (5x - 2) n (5x - 2) = 1/5 m hewa (5x - 2) + C.

E like me ka olelo hooholo ia mea maopopo ia no ka hana i papa integrand nele multiplier 5. E hookui aku ia i loko o laua me kēia hoonui loa ma 1/5 i mau 'aoʻao i ole hoʻololi i.

Hoʻohui 'ma ka māhele

E hoomanao i mau oihana - Z (y me) a me ka m (y me). Ka mea, pono e noke differentiable ma luna o kona aupuni. I kekahi differentiation waiwai mākou i: d (XZ) = xdz + zdx. Ka hoʻononiakahiʻana o nāʻaoʻaoʻelua, ua loaʻa: ∫d (XZ) = ∫ (xdz + zdx) => zx = ∫zdx + ∫xdz.

Rewriting ka kūpono helu kaulike, ua loaa i ka haʻilula, a wehewehe i ke ano o ka hoʻohui 'ma ka māhele: ∫zdx = zx - ∫xdz.

No ke aha la ka mea pono? Ka nui, e hoike i kekahi o na examples ia mea hiki ke Nohie, e ka olelo aku, e ho'ēmi i ∫zdx ∫xdz, ina ka hope, ua kokoke i ka tabular ano. Nō hoʻi, i kēia haʻilula hiki ke hoʻohana 'oi ma mua o kekahi manawa, no ka mea, e pono ai nā hualoaʻa.

Pehea e hoʻoponopono i indefinite integrals i kēia ala.

• pono e hoʻomaulia i ∫ (s + 1) E ^2s DS

∫ (m + ^{1) E 2s DS = {Z =} mau + 1, dz = DS, y me = 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;

• pono ho omaulia ∫lnsds

∫lnsds = {Z = lns, dz = DS / s, y me = s, dy = DS} = slns - ∫s m DS / s = slns - ∫ds = slns -s + C = mau (lns-1) + C.

Waiho hou i ka ee iaaanu aey

Keia rula o ka hoʻoponopono indefinite integrals i ole e emi i koi ma mua o ka mua elua, aole nae huikau. Ka papa hana Ua like penei: E V (m) - ka integral o kekahi papa V (m). I ka hanana i loko iho integral ma Laʻana slozhnosochinenny mai, hoʻoneʻe 'ia paha e kiʻi pioloke, a hele aku ia i ka hewa ala pāʻoihana. E pale ia me ka hana pono Lele ma ka ee iaaanu aey m e Z, ma a ka mau 'aoʻao o visually Nohie oiai e malama ana i ka Z ke kaumaha ma muli m.

I ka makemakika hua'ōlelo, keia mea like penei: ∫v (m) dx = ∫v (y me (Z)) y me '(Z) dz = V (Z) = V (y me -1 (x)), kahi m = y me ( Z) - hakahaka. A, o ka papa, i ka inverse papa Z = y me ^-1 (m) hoʻopau ai i ke ho'ākāka i ka pilina ma, a me ka pilina o aiaiiuo. Nui 'ana i kumumanaʻo - ka palena pau dx pono auou caiaiai me ka hou palena pau dz, mahope mai o ka loli ana o ee iaaanu aey i loko o ka indefinite integral komo ai ke kūapoʻana ia mai'ō a'ō,ʻaʻole pono i loko o ka integrand.

Eia kekahi laʻana:

• pono loaʻa ∫ (s + 1) / (s ² + 2s - 5) DS

Pili i ka hakahaka Z = (s + 1) / (s + ² 2s-5). A laila, dz = 2sds = 2 + 2 (s + 1) DS <=> (s + 1) DS = dz / 2. E like me ka hopena, i ka hahai ana ia olelo, i ka loa oluolu e hoʻomaulia:

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

• oe e huli i ka integral ∫2 ^mau E ^mau dx

E hoʻoponopono i ka rewrite i loko o ka palapala penei:

^{∫2 mau E mau DS = ∫ (} 2e) mau DS.

Mākou Kû ke ma kekahi = 2e (hope o ka manaʻo hoʻopiʻi kū'ē i kēia koe i mea ole, ka mea, ua nō mau) 'ōlelo, ua haawi aku mākou seemingly huikau integral i ka walaʻauʻana tabular ano:

^{∫ (2e) mau DS = ∫a} mau DS = he ka / lna + C = (2e) ke / Ln (2e) + C = 2 ^mau E ^mau / Ln (+ 2 lne) + C = 2 ^mau E ^mau / (ln2 + 1) + C.

Summing i ka palena pau hōʻailona

Ma a me ka nui, i kēia hana o ka indefinite integrals - ke kaikaina māhoe o ka loina o ka loli o ke ee iaaanu aey, akā, ma laila nō oko ao ma ka manao kaʻina. E mākou noʻonoʻo i hou au mamuli.

Inā ∫v (m) dx = V (m) + C a me y me = Z (m), laila ∫v (y me) dy = V (y me) + C.

I ka Ia manawa, ua pono i poina i ka AICIeOEOI integral loli ', i waena o i:

• dx = D (m + i), a kahi - kēlā ikaika;
• dx = (1 / he) D (koʻi + b), kahi he - a eia hou, akā,ʻaʻole Aʻohe;
• xdx = 1 / 2D (m ² + b);
• sinxdx = -d (cosx);
• cosxdx = D (sinx).

Inā ka kekahi mau hihia kahi a mākou e huli i ka indefinite integral mākou noonoo, examples hiki ke subsumed ma lalo o ka mau haʻilula makou '(m) dx = dw (m).

examples:

• pono loaʻa ∫ (2s + 3) ² DS, DS = 1 / 2D (2s + 3)

∫ (2s + 3) ² DS = 1 / 2∫ (2s + 3) ² b (2s + 3) = (1/2) m ((2s + 3) ²⁾ / 3 + C = (1/6) m (2s + 3) ² + C;

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

Online kōkua

Ma kekahi mau hihia, i ka hewa o ka i hiki lilo paha palaualelo, a me ka manaʻo pono, e hiki ke hoʻohana i ka online, hoʻopili, a aho, e hoʻohana i kekahi me ka m'kini helu indefinite integrals. I loko nō o ka nohoaliʻi ka manaʻo pohihihi a me controversial ano o na integrals, i ka olelo hooholo mea paha i kā lākou kekahi algorithm, i ua muli ma luna o ka loina hooponopono o ka "ina e hana ole ... laila ...".

O ka holo ana, He pakahi aku la ia pohihihi examples o ia i kekahi me ka m'kini helu eʻaʻole E ke Kumu, e like me ka mea i hihia i loko o a me ka hooholo ana i ka imi i ka artificially "puʻe wale akula kēlā" ma kekahi oihana mua i loko o ke kaʻina i hoʻolaunaʻia ma, no ka mea, i nā hualoaʻa e ahuwale aoao e hiki. I loko nō o ka controversial ano o keia olelo, ka mea, he oiaio, e like me ka makemakika, ma ka rula, he Hōʻuluʻulu 'epekema, a me kona iniiaiie Pahuhopu E hoomanao i ka mea pono, e Kuhina i ka palena. No ka, no ka mea, he ala apuupuu holo-ma na theories ka loa paʻakikī, e neʻe mai a me ka Hawaiʻi, no laila mai i kuhi i na ano he kumu hoʻohālike o ka hoʻoponopono indefinite integrals, i haawi mai - ia o ke kiekie o ka manawa kūpono. Akā hoʻi i kaʻaoʻao oaoieei-o na mea. Ma ka liʻiliʻi loa, e kaha i ka helu, e hiki ke hoʻohana i ka hana i loko o i ka mea i palapala ia iā mākou. Inā he mea i pono ai no ka 'akomi i ka ho omaulia ana i luna', ka 'aoʻao, a laila ka mea, e ole i ke akoakoa ai i ka oi nui lako polokalamu. E hoʻolohe kuhikuhi ma ke kaiapuni MatLab.

palapala noi

Na olelo hooholo o ka indefinite integrals ma mua kilohi pono loa Hoʻokaʻawale mai ka ike maopopo, no ka mea, he pa, e ike i ka maopopo ana o ka pelane. Ua, pono hoʻohana ia anywhere oe e hiki ole, akā, ka mea mau i ka pono waena hehee ai i loko o ke kaʻina hana o ka withdrawal o ka pāʻoihana i hoʻohana ma ka hoʻomaʻamaʻa. Ke'ī mai, i ka hoʻohui 'ia o hope differentiation, pela' Imi komo i loko o ke kaʻina hana o ka hoʻoponopono nā helu kaulike.
Ma hoʻi, i kēia mau nā helu kaulike i ka pololei hopena ma luna o ka hooholo ana o ÿano pilikia, ke kinoea make i ka ho omaulia a me thermal conductivity - ma pōkole, na mea a pau i kūpono ai ka makana, a shaping i ka wā e hiki mai. Indefinite integral, examples mākou no i i manaoia ma luna, AICIeOEOI wale nō i mua kilohi, me he kanaka hewa, e lawe mai hou, a nui hou discoveries.