Fa'ata'ita'iga o Fesili ma Talanoaga: Fa'amatalaga o le Indefinite Integral
O le integral lē mautinoa o se manatu autū i le calculus, e faʻaaogaina e suʻe ai le antiderivative o se galuega faatino ua tuʻuina atu. E taʻua foʻi o le antidifferentiation. I totonu o lenei tusiga, o le a tatou talanoaina ni faʻataʻitaʻiga o integral lē mautinoa, faʻatasi ai ma faʻamatalaga, ina ia malamalama atili ai i lenei manatu.
Malamalama i Fa'atulagaga Fa'avae e le Fa'atulagaina
O le integral lē mautinoa o le faagasologa lea o le sailia o le galuega muamua \( F(x) \) mai se derivative \( f(x) \), lea e faʻailoa mai e:
\[ \int f(x) \, dx = F(x) + C \]
lea o le \( C \) o se tumau taua. E tula'i mai lenei tumau ona o le fua o se tumau e leai se mea, o lea i le faagasologa o le anti-differentiation, e tatau ona tatou mafaufau i le avanoa e iai ai se tumau faapena.
Fua Fa'avae mo Fa'atulagaga e le Fa'atapula'aina
O nisi o fua fa'atatau masani e masani ona fa'aaogaina i integrals lē mautinoa e aofia ai:
1. \[ \int k \, dx = kx + C \]
lea o le \( k \) o se tumau.
2. \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
mo \( n \neq -1 \).
3. \[ \int e^x \, dx = e^x + C \]
4. \[ \int a^x \, dx = \frac{a^x}{\ln a} + C \]
lea o le \( a \) o se numera moni lelei ma le \( a \neq 1 \).
5. \[ \int \frac{1}{x} \, dx = \ln |x| +C\]
6. \[ \int \sin x \, dx = -\cos x + C \]
7. \[ \int \cos x \, dx = \sin x + C \]
Faʻataʻitaʻiga o Faʻafitauli Tau Faʻatasi e le Mautinoa ma a Latou Talanoaga
Faʻataʻitaʻiga 1
Fesili:
Fuafua le integral lē mautinoa o le \( f(x) = 3x^2 \).
Talanoaga:
Mo le foia o lenei integral, matou te faʻaaogaina le fua faʻavae o le integral mo galuega faatino o le foliga \( x^n \):
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
I lenei tulaga, o loʻo ia i tatou le \( f(x) = 3x^2 \), lea o le \( k = 3 \) ma le \( n = 2 \). Ona:
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{x^{3}}{3} \right) + C = x^3 + C \]
O lea la, \( \int 3x^2 \, dx = x^3 + C \).
Faʻataʻitaʻiga 2
Fesili:
Fuafua le fa'ai'uga le mautinoa o le \( f(x) = \frac{1}{x} \).
Talanoaga:
O le integral o le \( \frac{1}{x} \) e faʻavae i luga o le fua faʻavae e faapea:
\[ \int \frac{1}{x} \, dx = \ln |x| +C\]
O lea la, \( \int \frac{1}{x} \, dx = \ln |x| + C \).
Faʻataʻitaʻiga 3
Fesili:
Fuafua le fa'ai'uga le mautinoa o le \( f(x) = e^x \).
Talanoaga:
O le integral o le \( e^x \) e faʻavae i luga o le fua faʻavae e faapea:
\[ \int e^x \, dx = e^x + C \]
O lea la, \( \int e^x \, dx = e^x + C \).
Faʻataʻitaʻiga 4
Fesili:
Fuafua le fa'ai'uga le mautinoa o le \( \sin x \).
Talanoaga:
O le integral o le \( \sin x \) e faʻavae i luga o le fua faʻavae e:
\[ \int \sin x \, dx = -\cos x + C \]
O lea la, \( \int \sin x \, dx = -\cos x + C \).
Faʻataʻitaʻiga 5
Fesili:
Fuafua le fa'ai'uga le mautinoa o le \( \cos x \).
Talanoaga:
O le integral o le \( \cos x \) e faʻavae i luga o le fua faʻavae e:
\[ \int \cos x \, dx = \sin x + C \]
O lea la, \( \int \cos x \, dx = \sin x + C \).
Faʻataʻitaʻiga 6
Fesili:
Fuafua le integral lē mautinoa o le \( 5x^{-3} \).
Talanoaga:
Mo le foia o lenei integral, matou te faʻaaogaina le fua faʻavae o le integral mo galuega faatino o le foliga \( x^n \):
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
I lenei tulaga, o loʻo ia i tatou le \( f(x) = 5x^{-3} \), lea o le \( k = 5 \) ma le \( n = -3 \). Ona:
\[ \int 5x^{-3} \, dx = 5 \int x^{-3} \, dx = 5 \left( \frac{x^{-3+1}}{-3+1} \right) + C = 5 \left( \frac{x^{-2}}{-2} \right) + C = -\frac{5}{2} x^{-2} + C \]
O lea la, \( \int 5x^{-3} \, dx = -\frac{5}{2} x^{-2} + C \).
Faʻataʻitaʻiga 7
Fesili:
Fuafua le integral lē mautinoa o le \( 4e^{2x} \).
Talanoaga:
Mo le foia o lenei integral, e manaʻomia ona tatou faʻaaogaina le substance \( u \). Seʻi o tatou setiina \( u = 2x \) ina ia \( du = 2dx \), poʻo \( dx = \frac{du}{2} \).
\[ \int 4e^{2x} \, dx = 4 \int e^{u} \, \frac{du}{2} = 2 \int e^{u} \, du \]
O lea la, o le integral o le \( e^u \) o le \( e^u \):
\[ 2 \int e^u \, du = 2e^u + C \]
Toe foʻi i fesuiaʻiga muamua:
\[ 2e^u + C = 2e^{2x} + C \]
O lea la, \( \int 4e^{2x} \, dx = 2e^{2x} + C \).
Fa'aaogāina o Fa'atulagaga Fa'avae e le Fa'atapula'aina
E lautele le fa'aaogaina o integrals lē mautinoa i le saienisi ma le inisinia. Mo se fa'ata'ita'iga, i le fisiki, e fa'aaogaina integrals lē mautinoa e su'e ai le mamao e malaga ai se mea pe a iloa lona saoasaoa e tusa ai ma le taimi. I le tamaoaiga, e mafai ona fa'aaogaina integrals lē mautinoa e su'e ai le tau atoa po'o le polofiti pe a iloa le fua faatatau o le suiga o le tau po'o le polofiti i le iunite.
I'uga
O le integral lē mautinoa o se manatu tāua i le calculus, e faʻaaogaina e suʻe ai antiderivatives o functions. O le malamalama i fua faʻatatau eseese o integral faavae e matuā tāua tele i le foʻia o faʻafitauli e aofia ai integral lē mautinoa. Faatasi ai ma le lava o le faʻataʻitaʻi e faʻaaoga ai faʻataʻitaʻiga e pei o na o loʻo talanoaina i luga, e mafai ai e se tasi ona aʻoaʻoina le metotia o integral lē mautinoa. O le manatu o integral lē mautinoa e lē gata ina i ai le taua faʻavae ae faʻapea foʻi ma le aoga i vaega eseese o le saienisi.
