Fa'ata'ita'iga o Fesili Talanoaga Fa'atasi
O le integral o se manatu faavae i le calculus e lautele lona faʻaaogaina i matāʻupu eseese, e aofia ai le fisiki, inisinia, ma le tamaoaiga. O lenei tusiga o le a suʻesuʻeina ai faʻataʻitaʻiga eseese o faʻafitauli o le integral ma a latou fofo e maua ai se malamalamaga loloto.
1. Malamalamaaga Fa'avae o Fa'aputuga
I se faaupuga faigofie, o le integral o le fa'agaioiga fa'afeagai lea o se derivative. E lua ituaiga o integrals e masani ona talanoaina, o lona uiga:
– Fa'atulagaga Fa'avae e le Fa'amautuina: o se fa'atulagaga fa'avae lea e leai ni tapula'a pito i luga ma pito i lalo ma ua fa'ailoa mai e le ∫ f(x) dx.
– Fa'atulagaga Mautū: o se fa'atulagaga fa'avae lea e iai tapula'a pito i luga ma tapula'a pito i lalo ma ua fa'ailoa mai e le ∫[a,b] f(x) dx.
O le integral lē mautinoa e masani ona ta'ua o le anti-derivative, ma o le i'uga o le a aofia ai le tumau C ona o le meatotino o se derivative tumau e zero.
2. Faʻataʻitaʻiga o Faʻafitauli Tau Faʻatasi e le Mautinoa
Faʻataʻitaʻiga 1: Faʻatulagaga Faigofie e le Mautinoa
Fuafua le ∫ x^2 dx.
Talanoaga:
Ua tatou iloa o le tulafono faavae o le tuufaatasia mo le ∫ x^n dx o le (x^(n+1))/(n+1) + C, lea o le C o le tumau o le tuufaatasia.
Mo le integral o loʻo i luga, n = 2:
∫ x^2 dx = (x^(2+1))/(2+1) + C
= (x^3)/3 + C.
O lea la, o le iʻuga o le ∫ x^2 dx o le (x^3)/3 + C.
Faʻataʻitaʻiga 2: Faʻatasiga o Galuega Faʻatele
Fuafua ∫ e^x dx.
Talanoaga:
O le tulafono faavae mo le integral exponential ∫ e^x dx o le e^x + C.
O lea la, o le iʻuga o le ∫ e^x dx o le e^x + C.
3. Faʻataʻitaʻiga o Faʻafitauli Faʻapitoa o le Tautua
Faʻataʻitaʻiga 1: Faʻatulagaga Faigofie Mautinoa
Fuafua le ∫[1,3] x^2 dx.
Talanoaga:
Muamua, tatou te maua le anti-derivative o le x^2, o le (x^3)/3.
O lea la ua tatou sui ia tapulaʻa:
∫[1,3] x^2 dx = [(3^3)/3 – (1^3)/3]
= [27/3 – 1/3]
= [9 – 1/3]
= 8 + 2/3 po'o le 8.6667.
O lea la, o le iʻuga o le ∫[1,3] x^2 dx o le 26/3 poʻo le 8.6667.
Faʻataʻitaʻiga 2: Faʻatasiga e ala i le Suiga
Fuafua le ∫[0,2] (2x + 1) dx.
Talanoaga:
Muamua, tatou te sailia le antiderivative o le 2x + 1, o le x^2 + x. O lea la ua tatou sui nei tapula'a:
∫[0,2] (2x+1) dx = [(2^2 + 2) – (0^2 + 0)]
= [(4 + 2) – 0]
= 6.
O lea la, o le taunuuga o le ∫[0,2] (2x + 1) dx e 6.
4. Faʻataʻitaʻiga o Faʻafitauli Faʻatasi ma le Metotia Faʻavaega
O le integral fa'avaega o se metotia e fa'aaogaina pe a faigata ona fuafuaina sa'o le integral o le oloa o galuega faatino e lua. O le fua fa'atatau mo le integral fa'avaega e fa'apea:
∫ u dv = uv – ∫ v du
Fa'ata'ita'iga: Fa'avaega Fa'a-Trigonometric
Fuafua ∫ xe^x dx.
Talanoaga:
O iinei tatou te faʻaaogaina ai le metotia faʻavaega. Faʻapea o le u = x ma le dv = e^x dx. Ona du = dx ma le v = e^x.
E faʻavae i luga o le fua faʻatatau o le vaega faʻatasi:
∫ xe^x dx = xe^x – ∫ e^x dx
= xe^x – e^x + C
= e^x(x – 1) + C.
O lea la, o le taunuuga o le ∫ xe^x dx o le e^x(x – 1) + C.
5. Faʻataʻitaʻiga o Faʻafitauli o le Trigonometric Integral
Faʻataʻitaʻiga: Faʻatasiga o Galuega Faʻavae o le Trigonometric
Fuafua ∫ cos(x) dx.
Talanoaga:
O le tulafono autū o le tu'ufa'atasiga o le cos(x) o le sin(x) + C.
O lea la, o le taunuuga o le ∫ cos(x) dx o le sin(x) + C.
Faʻataʻitaʻiga: Faʻatasiga o Galuega Taulima ma Tapulaʻa
Fuafua le ∫[0,π/2] sin(x) dx.
Talanoaga:
Muamua, tatou te maua le anti-derivative o le sin(x), o le -cos(x).
Ia, sui ia tapula'a:
∫[0,π/2] agasala(x) dx = [ -cos(π/2) – (-cos(0)) ]
= [ -0 – (-1) ]
= 1.
O lea la, o le taunuuga o le ∫[0,π/2] sin(x) dx e 1.
6. Faʻataʻitaʻiga o le Faʻafitauli o le Suiga o le Integral
Faataitaiga: Suiga Fa'atasi
Fuafua ∫ 2x sqrt(1-x^2) dx.
Talanoaga:
Faaaoga le suiga u = 1-x^2, ona sosoo ai lea ma le du = -2x dx.
Ona suia lea o le vaega taua i le:
∫ sqrt(u) (-1/2 du)
= -1/2 ∫ u^(1/2) du
= -1/2 [ (2/3) u^(3/2) ] + C
= -1/3 (1-x^2)^(3/2) + C.
O lea la, o le taunuuga o le ∫ 2x sqrt(1-x^2) dx o le -1/3 (1-x^2)^(3/2) + C.
7. Faaiuga
O integrals o se meafaigaluega aoga tele i le matematika mo le sailia o le eria i lalo o se pi'o, voluma, ma le tele o isi fa'aoga. O le malamalama i metotia eseese o le integrations, e pei o le sui, partials, ma fa'avae o integrals, e taua tele. O fa'ata'ita'iga o lo'o talanoaina i luga o le a fa'amoemoe e fesoasoani e fa'amalosia ai lou malamalama i le manatu o integrals.
O le fa'ata'ita'i soo ma le malamalama i manatu fa'avae e taua tele i le tomai i le fa'atinoina o mea tu'ufa'atasi. Fa'aauau pea ona fa'ata'ita'i i fesuia'iga eseese ma fa'atulagaga fa'atino eseese e fa'alautele ai lou malamalama i lenei matā'upu.
Faʻamoemoe o le a aoga lenei tusiga mo oe i le aʻoaʻoina o integrals.
