Tsim, Secondary kev kawm ntawv thiab cov tsev kawm ntawv
Lub indefinite ib. Computation ntawm indefinite integrals
Ib tug ntawm cov yam tseem ceeb seem ntawm zauv tsom xam yog lub integral calculus. Nws npog ib tug heev dav teb ntawm cov khoom, uas tus thawj - nws yog ib lub indefinite ib. Txoj hauj lwm nws stands raws li ib tug tseem ceeb uas yog tseem nyob rau hauv high school qhia ib tug zuj zus tooj ntawm prospects thiab lub sij hawm, uas qhia txog kev kawm leb siab.
tsos
Thaum xub thawj siab ib muag, nws nkawd utterly ib rau niaj hnub, tshuaj pleev, tab sis nyob rau hauv kev xyaum nws hloov tawm hais tias nws tuaj rov qab rau hauv 1800 BC. Tsev rau officially xam tau tias yog tim lyiv teb chaws raws li tsis cuag peb ua ntej lawm pov thawj ntawm nws lub neej. Nws vim tsis muaj ntaub ntawv, tag nrho cov thaum positioned cias li ib tug tshwm sim. Nws ib zaug dua qhia rau cov theem ntawm scientific kev loj hlob ntawm cov neeg ntawm cov sij hawm. Thaum kawg, tej hauj lwm twb pom lub ancient Greek mathematicians, yos los ntawm lub xyoo pua 4 BC. Lawv piav qhia txog txoj kev siv qhov twg lub indefinite integral, lub essence ntawm cov uas yog mus nrhiav qhov ntim los yog cheeb tsam ntawm ib tug curvilinear zoo (peb-dimensional thiab ob-seem dav hlau, ntsig txog). muab xam tau raws li nyob rau hauv lub hauv paus ntsiab lus ntawm division ntawm tus thawj daim duab rau hauv infinitesimal Cheebtsam, muab hais tias lub ntim (cheeb tsam) yog twb paub rau lawv. Thaum lub sij hawm, txoj kev loj hlob, Archimedes siv nws mus nrhiav tau qhov chaw ntawm ib tug parabola. Zoo li suav nyob rau tib lub sij hawm rau kev ce nyob rau hauv ancient Tuam Tshoj, nyob qhov twg lawv twb kiag ywj siab los ntawm cov Greek khub science.
txoj kev loj hlob
Cov tom ntej no txhob nyob rau hauv lub XI xyoo pua BC tau ua tus ua hauj lwm ntawm lub as kawm "wagon" Abu Ali al-Basri, uas raug thawb rau ib thaj tsam ntawm lub twb paub, tau muab los ntawm cov ib mis rau xam cov qhaub ntawm tus nqi thiab degrees los ntawm tus thawj rau lub plaub, thov rau qhov no npe hu rau peb induction txoj kev.
Minds ntawm hnub no yog admired los ntawm lub ancient Egyptians tsim lub amazing monuments tsis muaj tshwj xeeb cov cuab yeej, tsuas yog rau cov uas ntawm lawv tus kheej txhais tes, tab sis yog tsis yog ib lub hwj chim chim zaum ntawm lub sij hawm tsis muaj tsawg ib tug txuj ci tseem ceeb? Muab piv nrog rau tam sim no lub sij hawm ntawm lawv lub neej zoo li luag txheej thaum ub, tab sis qhov kev txiav txim ntawm indefinite integrals deduced qhov txhia chaw thiab siv nyob rau hauv kev xyaum rau kev loj hlob ntxiv.
Cov kauj ruam tom ntej coj qhov chaw nyob rau hauv lub XVI xyoo pua, thaum lub Italian mathematician Cavalieri coj indivisible txoj kev, uas khaws Ib Ferma. Cov ob cwm pwm nteg lub lub hauv paus rau lub neej tshiab nimno integral calculus, uas yog lub npe hu thaum lub caij. Lawv khi lub tswv yim ntawm kev ntau yam zoo thiab koom ua ke, uas tau yav tas los pom raws li nws tus kheej-muaj chav nyob. By thiab loj, zauv ntawm lub sij hawm yog fragmented hais tshawb pom muaj nyob ntawm lawv tus kheej, tsis muaj ntau siv. Txoj kev sib sau ua ke thiab nrhiav tau ib qho av yog tus tsuas yog muaj tseeb nyob rau lub sij hawm ntawd, ua tsaug rau nws, cov niaj hnub zauv tsom xam muaj lub sij hawm kom loj hlob thiab tsim.
Nrog rau cov zaj uas lub sij hawm hloov txhua yam thiab lub ib lub cim zoo li. By thiab loj, nws twb xaiv zaum uas nyob rau hauv nws tus kheej txoj kev, piv txwv li, Newton siv ib tug square icon, uas muab tso rau ib tug integrable muaj nuj nqi, los yog tsuas yog muab tso ua ke.
Lub formal txhais
Lub indefinite ib nyob rau hauv lub ntsiab txhais ntawm lub txheej thaum ub, yog li peb xav txog nws nyob rau hauv thawj qhov chaw.
Antiderivative - yog rov qab muaj nuj nqi ntawm lub derivative, nyob rau hauv kev xyaum nws yog hu ua txheej thaum ub. Txwv tsis pub: txheej thaum ub muaj nuj nqi ntawm d - yog ib tug muaj nuj nqi D, uas yog lub derivative v <=> V '= v. Nrhiav txheej thaum ub yog los mus laij lub indefinite integral, thiab cov txheej txheem nws tus kheej yog hu ua kev koom ua ke.
Piv txwv li:
Cov kev ua s (y) = y 3, thiab nws cov txheej thaum ub S (y) = (y 4/4).
Lub teeb ntawm tag nrho cov primitives ntawm cov nuj nqi - qhov no yog ib qho indefinite integral, denoted nws raws li nram no: ∫v (x) dx.
Los ntawm virtue ntawm qhov tseeb hais tias V (x) - tsuas yog ib txheej thaum ub yuavtsum muaj nuj nqi, qhia tuas: ∫v (x) dx = V (x) + C, qhov twg C - qhov. Nyob rau hauv lub arbitrary qhov hais txog tej tsis tu ncua, txij thaum nws cov derivative yog pes tsawg.
zog
Cov khoom muaj los ntawm indefinite integral, yeej tseem zoo raws li nyob rau hauv lub ntsiab txhais thiab thaj chaw ntawm derivatives.
Xav txog cov ntsiab lus tseem ceeb:
- ib derivative ntawm lub txheej thaum ub yog txheej thaum ub nws tus kheej ntxiv rau arbitrary qhov C <=> ∫V '(x) dx = V (x) + C;
- derivative ntawm lub ib ntawm ib tug muaj nuj nqi yog tus thawj muaj nuj nqi <=> (∫v (x) dx) '= v (x);
- qhov no yog rho tawm los ntawm nyob rau hauv lub integral kos npe rau <=> ∫kv (x) dx = k∫v (x) dx, qhov twg k - yog arbitrary;
- integral, uas yog coj los ntawm cov sum ntawm cov identically sib npaug zos rau cov sum ntawm integrals <=> ∫ (v (y) + w (y)) dy = ∫v (y) dy + ∫w (y) dy.
Lub ob zog yuav xaus lus tias tus indefinite ib yog linear. Vim rau qhov no, peb tau: ∫ (kv (y) dy + ∫ LW (y)) dy = k∫v (y) dy + l∫w (y) dy.
Mus saib cov piv txwv ntawm kev kho cov ntsiab indefinite integrals.
Koj yuav tsum nrhiav tau cov ib ∫ (3sinx + 4cosx) dx:
- ∫ (3sinx + 4cosx) dx = ∫3sinxdx + ∫4cosxdx = 3∫sinxdx + 4∫cosxdx = 3 (-cosx) + 4sinx + C = 4sinx - 3cosx + C.
Los ntawm cov piv txwv peb yuav xaus uas koj tsis paub yuav ua li cas los daws kom tau indefinite integrals? Cia li nrhiav tau tag nrho cov primitives! Tab sis qhov kev tshawb rau cov hauv paus ntsiab lus sib tham hauv qab no.
Txoj kev thiab piv txwv
Nyob rau hauv thiaj li yuav daws tau qhov ib, koj muaj peev xwm chaw uasi mus rau nram qab no txoj kev:
- npaj txhij mus rau noj kom zoo dua ntawm lub rooj;
- ua ke los ntawm qhov chaw;
- kev los ntawm hloov lub nce mus nce los;
- summing up nyob rau hauv lub kos npe rau ntawm lub differential.
ntxhuav
Cov feem ntau yooj yim thiab txoj kev zoo. Thaum lub caij, zauv tsom xam yuav khav heev uas nws kim heev lub rooj, uas sau tawm cov yooj yim mis ntawm indefinite integrals. Nyob rau hauv lwm yam lus, muaj templates muab rau koj thiab koj tsuas coj kom zoo dua ntawm lawv. Ntawm no yog daim ntawv teev cov ntsiab lus txaus qhia, uas yuav muab tso tawm kom txhua Piv txwv li, muaj ib cov tshuaj:
- ∫0dy = C, qhov twg C - qhov;
- ∫dy = y + C, qhov twg C - qhov;
- ∫y n dy = (y n + 1) / (n + 1) + C, qhov twg C - ib tug tsis tu ncua, thiab n - xov tooj sib txawv los ntawm kev sib sau;
- ∫ (1 / y) dy = ln | y | + C, qhov twg C - qhov;
- ∫e y dy = e y + C , qhov twg C - qhov;
- ∫k y dy = (k y / haujlwm uas yog COy k) + C, qhov twg C - qhov;
- ∫cosydy = siny + C, qhov twg C - qhov;
- ∫sinydy = -cosy + C, qhov twg C - qhov;
- ∫dy / cos 2 y = tgy + C, qhov twg C - qhov;
- ∫dy / txoj kev txhaum 2 y = -ctgy + C, qhov twg C - qhov;
- ∫dy / (1 + y 2) = arctgy + C, qhov twg C - qhov;
- ∫chydy = paub txaj muag + C, qhov twg C - qhov;
- ∫shydy = Chy + C, qhov twg C - qhov.
Yog hais tias tsim nyog, ua ib tug ob peb ntawm cov kauj ruam ua integrand mus rau ib tug tabular saib thiab txaus siab rau lub yeej. QAUV: ∫cos (5x -2) dx = 1 / 5∫cos (5x - 2) d (5x - 2) = 1/5 x kev txhaum (5x - 2) + C.
Raws li qhov kev txiav txim nws yog tseeb hais tias piv txwv li ib cov lus integrand lacks d 5. Peb ntxiv nws nyob rau hauv parallel nrog no txawj los ntawm 1/5 rau general qhia tsis hloov.
Kev koom ua ke los ntawm qhov chaw
Xav txog ob zog - z (y) thiab x (y). Lawv yuav tsum tsis tu ncua differentiable rau nws sau. Nyob rau hauv ib ntau yam zog peb muaj: d (xz) = xdz + zdx. Ua ke ob sab, peb tau txais: ∫d (xz) = ∫ (xdz + zdx) => ZX = ∫zdx + ∫xdz.
Rewriting lub resulting kab zauv, peb tau txais cov mis, uas qhia txog tus qauv ntawm kev koom ua ke los ntawm qhov chaw: ∫zdx = ZX - ∫xdz.
Yog vim li cas yog nws tsim nyog? Lub fact tias ib txhia ntawm cov piv txwv nws yog tau mus piv rau, cia hais tias, kom tsis txhob muaj ∫zdx ∫xdz, yog hais tias tus tom kawg yog nyob ze rau lub tabular daim ntawv. Tsis tas li ntawd, qhov no mis yuav siv ntau tshaj li ib zaug, kom pom tau.
Yuav ua li cas los daws kom tau indefinite integrals no txoj kev:
- tsim nyog los xam ∫ (s + 1) e 2s ds
∫ (x + 1) e 2s ds = {z = s + 1, dz = ds, y = 1 / 2e 2s, dy = e 2 x ds} = ((s + 1) e 2s) / 2-1 / 2 ∫e 2s dx = ((s + 1) e 2s) / 2-e 2s / 4 + C;
- yuav tsum xam ∫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.
Hloov cov kuj sib txawv
Qhov no txoj ntsiab cai ntawm kev daws indefinite integrals yog tsis tsawg nyob rau hauv coob tshaj yav dhau los ob tug, tab sis yog tham. Cov qauv no yog raws li nram no: Cia V (x) - lub ib ntawm ib co muaj nuj nqi v (x). Nyob rau hauv cov kev tshwm sim hais tias nyob rau hauv nws tus kheej ib nyob rau hauv Piv txwv li slozhnosochinenny los, yog yuav tau tsis meej pem thiab mus down tsis ncaj ncees lawm txoj kev daws teeb meem. Kom tsis txhob xyaum ua li no hloov los ntawm lub nce mus nce los x rau z, nyob rau hauv uas cov kev qhia visually yooj yim zog thaum tswj cov z nyob ntawm x.
Nyob rau hauv lej lus, qhov no yog raws li nram no: ∫v (x) dx = ∫v (y (z)) y '(z) dz = V (z) = V (y -1 (x)), qhov chaw uas x = y ( z) - hloov chaw. Thiab, ntawm chav kawm, cov ntxeev ua z = y -1 (x) tag nrho qhia txog cov kev sib raug zoo thiab cov kev sib raug zoo ntawm cov tsiaj ntawv. Tseem ceeb daim ntawv - lub differential dx tas hloov nrog ib tug tshiab differential dz, txij li thaum cov kev hloov ntawm nce mus nce los nyob rau hauv lub indefinite ib yuav hloov nws qhov txhia chaw, tsis yog nyob rau hauv lub integrand.
Piv txwv li:
- yuav tsum nrhiav ∫ (s + 1) / (s 2 + 2s - 5) ds
Thov lub hloov z = (s + 1) / (s 2 + 2s-5). Ces dz = 2sds = 2 + 2 (s + 1) ds <=> (s + 1) ds = dz / 2. Raws li ib tug tshwm sim, lub nram qab no qhia, uas yog heev yooj yim mus xam:
∫ (s + 1) / (s 2 + 2s-5) ds = ∫ (dz / 2) / z = 1 / 2ln | z | + C = 1 / 2ln | s 2 + 2s-5 | + C;
- koj yuav tsum nrhiav tau cov ib ∫2 s e s dx
Yuav kom daws tau cov sau dua tshiab nyob rau hauv cov nram no daim ntawv:
∫2 s e s ds = ∫ ( 2e) s ds.
Peb txhais los ntawm ib tug = 2e (kev hloov ntawm lub cav kauj ruam no yog tsis yog, nws yog tseem s), peb muab peb cov seemingly nyuab ib txog kev sau tabular daim ntawv:
∫ (2e) s ds = ∫a s ds = ib s / lna + C = (2e) s / haujlwm uas yog COy (2e) + C = 2 s e s / haujlwm uas yog COy (2 + lne) + C = 2 s e s / (ln2 + 1) + C.
Summing li ib tug kos npe rau differential
By thiab loj, no txoj kev ntawm indefinite integrals - tus ntxaib tij laug ntawm tus hauv paus ntsiab lus ntawm cov kev hloov ntawm nce mus nce los, tab sis muaj cov sib txawv nyob rau hauv tus txheej txheem ntawm kev cuv npe. Cia peb xav txog nyob rau hauv ntau yam.
Yog hais tias ∫v (x) dx = V (x) + C thiab y = z (x), ces ∫v (y) dy = V (y) + C.
Nyob rau tib lub sij hawm peb yuav tsum tsis txhob hnov qab cov tseem ceeb ib transformations, cov uas:
- dx = d (x + a), thiab nyob rau - txhua qhov;
- dx = (1 / a) d (taus + b), qhov chaw uas ib tug - qhov dua, tab sis tsis pes tsawg;
- xdx = 1 / 2d (x 2 + b);
- sinxdx = -D (cosx);
- cosxdx = d (sinx).
Yog hais tias peb xav txog cov kev cov ntaub ntawv uas peb xam cov indefinite integral, piv txwv yuav subsumed nyob rau hauv lub dav dav mis w '(x) dx = DW (x).
piv txwv:
- yuav tsum nrhiav ∫ (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 pab
Nyob rau hauv tej rooj plaub, lub txhaum ntawm cov uas muaj peev xwm ua los yog kev tub nkeeg, los yog ib qho mob ceev yuav tsum tau, koj muaj peev xwm siv lub online ras, los yog es, siv ib tug xam zauv indefinite integrals. Dua li ntawm qhov khees complexity thiab muaj teeb meem xwm ntawm lub integrals, qhov kev txiav txim yog kev kawm mus rau lawv cov algorithm, uas yog raws li nyob rau hauv lub hauv paus ntsiab lus ntawm "yog hais tias koj ua tsis tau ... ces ...".
Ntawm cov hoob kawm, ib tug tshwj xeeb intricate piv txwv ntawm xws li ib tug xam zauv yuav tsis tswv, raws li muaj mob nyob rau hauv uas ib tug txiav txim tau mus nrhiav tau ib lub artificially "yuam" los ntawm qhia tej ntsiab nyob rau hauv tus txheej txheem, vim hais tias cov kev tshwm sim yog cuab kev txoj kev mus cuag. Txawm tias cov muaj teeb meem xwm ntawm nqe lus no, nws yeej muaj tseeb, raws li cov kev kawm txog zauv, nyob rau hauv tus, paub daws teeb science, thiab nws cov thawj lub hom phiaj considers qhov yuav tsum tau los txhawb tus ciam teb. Tseeb, rau ib tug tus mus dhia-nyob rau hauv lub theories yog heev yooj yim mus txav mus thiab evolve, yog li ntawd tsis txhob xav tias cov piv txwv ntawm kev daws indefinite integrals, uas tau muab rau peb - qhov no yog qhov siab ntawm lub sij hawm. Tab sis rov qab mus rau sab kev ntawm tej yam. Yam tsawg kawg mus xyuas cov lus teb, koj yuav siv tau cov kev pab cuam nyob rau hauv uas nws tau sau ntawv mus rau peb. Yog hais tias muaj yog ib tug yuav tsum tau rau automatic muab xam los ntawm txoj kab zauv, ces lawv tsis muaj mus rau chaw uasi mus rau ib tug ntau loj software. Yuav tsum xyuam xim feem ntau nyob rau hauv lub cheeb tsam matlab.
daim ntawv thov
Qhov kev txiav txim ntawm indefinite integrals tom ntej glance nkawd kiag li detached ntawm kev muaj tiag, vim hais tias nws yog ib qhov nyuaj rau pom cov cuab siv ntawm lub dav hlau. Tseeb, ncaj qha siv lawv nyob qhov twg koj tsis tau, tab sis lawv yog ib tug tsim nyog intermediate caij nyob rau hauv tus txheej txheem ntawm tshem ntawm kev daws siv nyob rau hauv xyaum. Yog li, kev koom ua ke ntawm rov qab ntau yam zoo, yog li koom tes nyob rau hauv tus txheej txheem ntawm kev daws sib npaug.
Nyob rau hauv lem, cov equations muaj ib tug ncaj qha feem rau qhov kev txiav txim ntawm cov neeg kho tshuab cov teeb meem, trajectory xam thiab thermal conductivity - nyob rau hauv luv luv, txhua yam uas tshwm sim tawm tam sim no thiab shaping lub neej yav tom ntej. Indefinite integral, piv txwv uas peb tau kawm saum toj no, tsuas tsis tseem ceeb thaum xub thawj siab ib muag, raws li ib puag mus nqa tawm ntau thiab ntau tshiab discoveries.
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