Ib qho ntawm cov ntsiab lus ntawm kev txheeb xyuas lej yog ib qho kev suav lej. Nws npog qhov dav tshaj plaws ntawm cov khoom, qhov twg thawj zaug yog qhov tsis muaj qhov tsis muaj tseeb. Nws yog ib qho tsim nyog muab nws ua tus yuam sij, uas txawm nyob hauv tsev kawm theem siab nthuav tawm ntau qhov kev xav thiab cov cib fim uas piav qhia txog lej siab dua.
Zoo
Thaum saib, qhov tseem ceeb zoo li niaj hnub, cuam tshuam, tab sis hauv kev xyaum nws hloov tawm tias nws tshwm sim thaum ntxov li 1800 BC. Tim lyiv teb chaws raug suav hais tias yog lub tebchaws, txij li cov pov thawj ua ntej ntawm nws lub neej tsis tau mus txog peb. Nws, vim tsis muaj ntaub ntawv, tag nrho lub sij hawm no tau positioned yooj yim raws li ib tug tshwm sim. Nws ib zaug tau lees paub txog theem ntawm kev loj hlob ntawm kev tshawb fawb ntawm cov neeg ntawm lub sijhawm ntawd. Thaum kawg, cov hauj lwm ntawm cov lej Greek thaum ub rov qab mus rau xyoo pua 4 BC tau pom. Lawv tau piav qhia txog ib txoj hauv kev uas ib qho tsis muaj kev sib koom ua ke tau siv, qhov tseem ceeb ntawm qhov uas yog nrhiav qhov ntim lossis thaj tsam ntawm daim duab curvilinear (peb-dimension althiab ob-dimensional dav hlau, ntsig txog). Lub hauv paus ntsiab lus suav yog ua raws li kev faib cov thawj daim duab rau hauv cov khoom tsis kawg, muab tias lawv qhov ntim (chaw) twb paub lawm. Sij hawm dhau mus, txoj kev tau loj hlob, Archimedes siv nws los nrhiav thaj tsam ntawm parabola. Cov kev suav zoo sib xws tau ua nyob rau tib lub sijhawm los ntawm cov kws tshawb fawb hauv Suav teb thaum ub, thiab lawv muaj kev ywj pheej ntawm lawv cov Greek cov koom nrog hauv kev tshawb fawb.
Kev txhim kho
Kev tawg tom ntej hauv xyoo pua 11 AD yog kev ua haujlwm ntawm Arab kws tshawb fawb- "universal" Abu Ali al-Basri, uas thawb cov ciam teb ntawm qhov uas twb paub lawm, muab cov qauv raws li qhov tseem ceeb rau kev suav cov lej. ntawm kab thiab cov sums ntawm powers los ntawm thawj mus rau plaub, thov rau qhov no txoj kev ntawm lej induction paub rau peb.
Lub siab ntawm lub sijhawm niaj hnub qhuas tias cov neeg Iyiv thaum ub tau tsim cov vaj tsev zoo nkauj heev uas tsis muaj cov cuab yeej tshwj xeeb, tshwj tsis yog lawv txhais tes, tab sis tsis yog lub zog ntawm lub siab ntawm cov kws tshawb fawb ntawm lub sijhawm ntawd tsis muaj qhov txuj ci tseem ceeb? Muab piv rau niaj hnub no, lawv lub neej zoo li yuav luag qub, tab sis kev daws teeb meem ntawm kev sib koom ua ke tau muab los ntawm txhua qhov chaw thiab siv hauv kev xyaum rau kev txhim kho ntxiv.
Cov kauj ruam tom ntej tau tshwm sim nyob rau hauv lub xyoo pua 16th, thaum tus lej Italian Cavalieri tau tsim cov txheej txheem ntawm indivisibles, uas tau khaws los ntawm Pierre Fermat. Nws yog ob tus cwj pwm no uas tau tsim lub hauv paus rau kev suav niaj hnub no, uas paub txog tam sim no. Lawv txuas cov ntsiab lus ntawm kev sib txawv thiab kev sib koom ua ke, uas yog yav tas loskho raws li autonomous units. Los ntawm thiab loj, cov lej ntawm lub sijhawm ntawd tau tawg, cov lus ntawm cov lus xaus muaj nyob ntawm lawv tus kheej, muaj qhov txwv. Txoj kev ntawm kev sib koom ua ke thiab tshawb nrhiav hauv av yog tib qhov tseeb nyob rau lub sijhawm ntawd, ua tsaug rau qhov kev tshawb fawb lej niaj hnub tau txais lub sijhawm los loj hlob thiab txhim kho.
Txhua yam tau hloov pauv raws sijhawm, suav nrog cov cim ntawm qhov tseem ceeb. Los ntawm thiab loj, cov kws tshawb fawb tau qhia nws los ntawm tag nrho cov txhais tau tias, piv txwv li, Newton siv lub cim square uas nws muab tso rau hauv ib qho kev sib koom ua ke lossis tsuas yog muab tso rau ib sab ntawm nws.
Qhov kev tsis sib haum xeeb no txuas ntxiv mus txog rau xyoo pua 17th, thaum tus kws tshawb fawb Gottfried Leibniz, lub cim tseem ceeb rau tag nrho txoj kev xav ntawm kev ua lej, qhia lub cim kom paub txog peb. Lub elongated "S" yog qhov tseeb raws li tsab ntawv no ntawm cov tsiaj ntawv Latin, raws li nws qhia txog cov txiaj ntsig ntawm cov tshuaj tiv thaiv. Qhov tseem ceeb tau txais nws lub npe ua tsaug rau Jacob Bernoulli 15 xyoo tom qab.
Formal definition
Lub indefinite integral ncaj qha nyob ntawm lub ntsiab txhais ntawm antiderivative, yog li cia peb xav txog nws ua ntej.
Ib qho tshuaj tiv thaiv yog ib txoj haujlwm uas yog qhov hloov pauv ntawm qhov sib txawv, hauv kev xyaum nws kuj hu ua primitive. Txwv tsis pub: tus antiderivative ntawm ib tug muaj nuj nqi d yog ib tug muaj nuj nqi D uas nws derivative yog sib npaug rau v V'=v. Kev tshawb nrhiav cov tshuaj tiv thaiv yog kev suav ntawm qhov tsis muaj qhov tsis sib xws, thiab cov txheej txheem no nws tus kheej hu ua kev sib koom ua ke.
Example:
Function s(y)=y3, and its antiderivative S(y)=(y4/4).
Cov txheej txheem ntawm tag nrho cov antiderivatives ntawm lub luag haujlwm hauv kev txiav txim siab yog qhov tsis muaj qhov tsis muaj tseeb, nws tau qhia raws li hauv qab no: ∫v(x)dx.
Vim qhov tseeb tias V(x) tsuas yog qee qhov kev tiv thaiv ntawm qhov kev ua haujlwm qub, qhov kev qhia tshwm sim: ∫v(x)dx=V(x) + C, qhov C yog qhov tsis tu ncua. Ib qho arbitrary tas li yog ib qho tas mus li, txij li nws cov txiaj ntsig yog sib npaug rau xoom.
Properties
Cov khoom uas qhov tsis muaj qhov tsis muaj tseeb muaj yog raws li lub ntsiab lus txhais thiab cov khoom ntawm derivatives.
Cia peb saib cov ntsiab lus tseem ceeb:
- qhov tseem ceeb los ntawm cov txiaj ntsig ntawm cov tshuaj tiv thaiv yog cov tshuaj tiv thaiv nws tus kheej ntxiv rau qhov tsis txaus ntseeg С ∫V'(x)dx=V(x) + C;
- tus derivative ntawm lub function integral yog thawj muaj nuj nqi (∫v(x)dx)'=v(x);
- constant raug rho tawm ntawm hauv qab lub cim kos npe ∫kv(x)dx=k∫v(x)dx, qhov twg k yog arbitrary;
- qhov sib piv tau los ntawm qhov sib npaug yog sib npaug rau cov lej ntawm cov integrals ∫(v(y) + w(y))dy=∫v(y)dy +∫w(y)dy.
Los ntawm ob lub zog kawg, peb tuaj yeem xaus tias qhov tsis muaj qhov tsis muaj tseeb yog kab. Ua tsaug rau qhov no, peb muaj: ∫(kv(y)dy +∫ lw(y))dy=k∫v(y)dy + l∫w(y)dy.
Yuav kom sib sau ua ke, xav txog cov piv txwv ntawm kev daws qhov tsis sib xws.
Nws yog qhov tsim nyog los nrhiav qhov tseem ceeb ∫(3sinx + 4cosx)dx:
∫(3sinx + 4cosx)dx=∫3sinxdx + ∫4cosxdx=3∫sinxdx + 4∫cosxdx=3(-cosx) + 4sinx + C=4sinx - 3cosxC
Los ntawm qhov piv txwv peb tuaj yeem xaus:tsis paub yuav ua li cas los daws qhov integras? Tsuas pom tag nrho cov primitives! Tab sis cov hauv paus ntsiab lus ntawm kev tshawb nrhiav yuav raug txiav txim siab hauv qab no.
Txoj kev thiab piv txwv
Yuav daws qhov tsis sib xws, koj tuaj yeem siv cov hauv qab no:
- siv lub rooj npaj;
- kev koom ua ke los ntawm ntu;
- kev koom ua ke los ntawm kev hloov pauv;
- ua rau hauv qab qhov sib txawv.
Tables
Txoj kev yooj yim tshaj plaws thiab txaus siab tshaj plaws. Nyob rau lub sijhawm no, kev txheeb xyuas lej suav nrog cov lus dav dav uas cov qauv ntawm cov lej tsis raug suav sau tau sau. Hauv lwm lo lus, muaj cov qauv uas tau tsim ua ntej koj thiab rau koj, nws tsuas yog siv lawv xwb. Nov yog cov npe ntawm cov lus tseem ceeb uas koj tuaj yeem muab tau yuav luag txhua qhov piv txwv uas muaj kev daws teeb meem:
- ∫0dy=C, qhov C yog qhov tsis tu ncua;
- ∫dy=y + C, qhov C yog qhov tsis tu ncua;
- ∫y dy=(yn+1) / (n + 1) + C, qhov C yog qhov tsis tu ncua thiab n - tsis yog tus lej;
- ∫(1/y)dy=ln|y| + C, qhov twg C yog qhov tas li;
- ∫eydy=ey + C, qhov twg C yog qhov tas li;
- ∫kydy=(ky/ln k) + C, qhov twg C yog qhov tas li;
- ∫cosydy=siny + C, qhov C yog qhov tas li;
- ∫sinydy=-cosy + C, qhov twg C yog qhov tas li;
- ∫dy/cos2y=tgy + C, qhov twg C yog qhov tas li;
- ∫dy/sin2y=-ctgy + C, qhov C yog qhov tas li;
- ∫dy/(1 + y2)=arctgy + C, qhov twg C yog qhov tas li;
- ∫chydy=txaj muag + C, qhov twg C -tas li;
- ∫shydy=chy + C, qhov C yog qhov tas mus li.
Yog tias tsim nyog, ua ob peb kauj ruam, nqa qhov sib koom ua ke rau hauv daim ntawv teev npe thiab txaus siab rau kev yeej. Example: ∫cos(5x -2)dx=1/5∫cos(5x - 2)d(5x - 2)=1/5 x sin(5x - 2) + C.
Raws li kev daws teeb meem, nws yog qhov tseeb tias rau qhov piv txwv tabular, qhov kev sib koom ua ke tsis muaj qhov tseem ceeb ntawm 5. Peb ntxiv nws, muab nws sib npaug los ntawm 1/5 nyob rau hauv parallel kom cov lus qhia dav dav tsis hloov.
Kev sib koom ua ke los ntawm ntu
Xav txog ob txoj haujlwm - z(y) thiab x(y). Lawv yuav tsum tsis tu ncua sib txawv ntawm tag nrho cov ntsiab lus. Raws li ib qho ntawm cov khoom sib txawv, peb muaj: d(xz)=xdz + zdx. Integrating ob qhov chaw ntawm kab zauv, peb tau: ∫d(xz)=∫(xdz + zdx)=> zx=∫zdx + ∫xdz.
Rewriting the resulting equality, peb tau ib tug formula uas piav txog cov txheej txheem ntawm kev koom ua ke los ntawm qhov chaw: ∫zdx=zx - ∫xdz.
Vim li cas thiaj xav tau? Lub ntsiab lus yog tias qee qhov piv txwv tuaj yeem ua kom yooj yim, hais lus, txo ∫zdx rau ∫xdz yog tias tom kawg yog ze rau daim ntawv tabular. Tsis tas li, cov qauv no tuaj yeem siv ntau dua ib zaug, ua tiav cov txiaj ntsig zoo.
Yuav ua li cas daws qhov tsis muaj qhov tsis muaj tseeb ntawm no:
yuav tsum xam ∫(s + 1)e2sds
∫(x + 1)e2sds={z=s+1, dz=ds, y=1/2e2s, dy=e2xds}=((s+1)e2s) / 2-1/2∫e2s dx=((s+1)e2s) / 2-e2s/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.
Variable hloov pauv
Lub hauv paus ntsiab lus ntawm kev daws qhov tsis muaj qhov tsis sib xws tsis muaj qhov xav tau tsawg dua li ob qho dhau los, txawm tias nws nyuaj dua. Cov txheej txheem yog raws li nram no: cia V(x) ua ib feem ntawm ib co muaj nuj nqi v(x). Nyob rau hauv cov kev tshwm sim hais tias lub integral nws tus kheej nyob rau hauv cov piv txwv los nyob rau hauv raws li complex, muaj ib tug high probability ntawm tsis meej pem thiab coj txoj kev tsis ncaj ncees lawm kev daws teeb meem. Txhawm rau zam qhov no, kev hloov pauv ntawm qhov sib txawv x mus rau z yog xyaum, uas qhov kev qhia dav dav pom tau yooj yim thaum tswj hwm qhov kev cia siab ntawm z ntawm x.
Mathematically it looks like this: ∫v(x)dx=∫v(y(z))y'(z)dz=V(z)=V(y-1(x)), qhov twg x=y(z) yog hloov pauv. Thiab, ntawm chav kawm, qhov inverse function z=y-1(x) qhia tag nrho txog kev vam khom thiab kev sib raug zoo ntawm kev sib txawv. Cov lus ceeb toom tseem ceeb - qhov sib txawv dx yuav tsum tau hloov pauv los ntawm qhov sib txawv dz tshiab, txij li kev hloov pauv ntawm qhov sib txawv hauv qhov tsis muaj qhov tsis muaj tseeb txhais tau tias nws hloov pauv txhua qhov chaw, thiab tsis yog hauv kev sib koom ua ke xwb.
Example:
xav nrhiav ∫(s + 1) / (s2 + 2s - 5)ds
Tau qhov hloov pauv z=(s+1)/(s2+2s-5). Ces dz=2sds=2+2(s+1)ds(s+1)ds=dz/2. Raws li qhov tshwm sim, peb tau txais cov lus qhia hauv qab no, uas yooj yim rau kev suav:
∫(s+1)/(s2+2s-5)ds=∫(dz/2)/z=1/2ln|z|+C=1/2ln|s2+2s-5|+C;
yuav tsum nrhiav qhov tseem ceeb∫2sesdx
Yuav daws, peb rov sau cov lus qhia hauv daim ntawv hauv qab no:
∫2sesds=∫(2e)sds.
Denote los ntawm a=2e (cov kauj ruam no tsis yog hloov pauv rau qhov kev sib cav, nws tseem yog s), peb coj peb qhov zoo li qhov sib xyaw ua ke mus rau hauv daim ntawv qhia theem qis:
∫(2e)sds=∫asds=as / lna + C=(2e)s / ln(2e) + C=2ses / ln(2 + lne) + C=2ses / (ln2 + 1) + C.
Nco hauv qab qhov sib txawv
Los ntawm qhov loj, hom kev sib xyaw ua ke no yog ib tus kwv tij ntxaib ntawm txoj cai hloov pauv hloov pauv, tab sis muaj qhov sib txawv ntawm cov txheej txheem tsim. Cia peb saib ntxiv.
If ∫v(x)dx=V(x) + C and y=z(x), then ∫v(y)dy=V(y) + C.
Hauv qhov no, ib tus yuav tsum tsis txhob hnov qab txog qhov tsis tseem ceeb ntawm kev hloov pauv, cov uas:
- dx=d(x + a), qhov twg a yog qhov tsis tu ncua;
- dx=(1 / a)d(ax + b), qhov twg a rov yog qhov qub, tab sis tsis sib npaug rau xoom;
- xdx=1/2d(x2 + b);
- sinxdx=-d(cosx);
- cosxdx=d(sinx).
Yog tias peb xav txog cov ntaub ntawv dav dav thaum peb suav qhov tsis muaj qhov tsis sib xws, cov piv txwv tuaj yeem suav nrog hauv cov qauv dav dav w'(x)dx=dw(x).
Examples:
xav nrhiav ∫(2s + 3)2ds, ds=1/2d(2s + 3)
∫(2s + 3)2ds=1/2∫(2s + 3)2d(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.
Kev Pab Online
Qee qhov xwm txheej, qhov txhaum uas yuav yog kev tub nkeeg lossis kev xav tau ceev, koj tuaj yeem siv cov lus qhia hauv online, lossis siv lub laij lej suav tsis txheeb. Txawm hais tias tag nrho qhov pom tseeb tsis yooj yim thiab tsis sib haum xeeb ntawm kev sib koom ua ke, lawv cov kev daws teeb meem yog raug rau qee qhov algorithm, uas yog raws li lub hauv paus ntsiab lus "yog tias tsis yog …, ces …".
Tau kawg, xws li lub laij lej yuav tsis paub tshwj xeeb tshaj yog cov piv txwv zoo sib xws, txij li muaj qee qhov kev daws teeb meem yuav tsum tau pom, "yuav yuam" qhia qee yam hauv cov txheej txheem, vim tias qhov tshwm sim tsis tuaj yeem ua tiav hauv qhov pom tseeb. txoj kev. Txawm hais tias txhua qhov kev tsis sib haum xeeb ntawm cov lus no, nws yog qhov tseeb, txij li kev ua lej, hauv paus ntsiab lus, yog kev tshawb fawb paub daws teeb meem, thiab txiav txim siab qhov yuav tsum tau nthuav dav thaj tsam ntawm qhov muaj peev xwm ua nws txoj haujlwm tseem ceeb. Tseeb tiag, nws yog qhov nyuaj heev rau kev txav mus los thiab txhim kho raws li kev sib tw, khiav-hauv txoj kev xav, yog li koj yuav tsum tsis txhob xav tias cov piv txwv ntawm kev daws qhov tsis sib xws uas peb tau muab yog qhov siab ntawm qhov muaj peev xwm. Tab sis rov qab mus rau sab technical ntawm yam. Tsawg kawg yog txhawm rau txheeb xyuas cov lej, koj tuaj yeem siv cov kev pabcuam uas txhua yam tau sau ua ntej peb. Yog tias muaj kev xav tau rau kev suav tsis siv neeg ntawm cov lus qhia nyuaj, tom qab ntawd lawv tsis tuaj yeem muab faib nrog, koj yuav tau mus rau software loj dua. Nws yog ib qho tsim nyog ua tib zoo saib ua ntej ntawm tag nrho rau MatLab ib puag ncig.
Daim ntawv thov
Cov kev daws teeb meem ntawm qhov tsis muaj qhov tsis sib xws thaum xub thawj siab ib muag zoo nkaus li tawm ntawm kev kov nrog qhov tseeb, vim nws nyuaj rau pom qhov pom tseeb ntawm daim ntawv thov. Tseeb tiag, lawv tsis tuaj yeem siv ncaj qha rau txhua qhov chaw, tab sis lawv suav tias yog qhov tsim nyog nruab nrab ntawm cov txheej txheem ntawm kev muab cov kev daws teeb meem siv hauv kev xyaum. Yog li, kev sib koom ua ke yog hloov pauv mus rau qhov sib txawv, vim nws koom nrog hauv cov txheej txheem ntawm kev daws qhov sib npaug.
Thiab, cov kev sib npaug no muaj kev cuam tshuam ncaj qha rau kev daws teeb meem ntawm kev siv tshuab, kev suav cov trajectories thiab thermal conductivity - luv luv, txhua yam uas ua rau tam sim no thiab ua rau yav tom ntej. Qhov tsis muaj tseeb, cov piv txwv uas peb tau tshuaj xyuas saum toj no, yog qhov tsis tseem ceeb tsuas yog thaum xub thawj siab ib muag, vim nws yog lub hauv paus rau kev tshawb pom ntau thiab ntau dua tshiab.
