Yuav Ua Li Cas Thiaj Daws Tau Partial Integrals: Ib Phau Ntawv Qhia Tag Nrho
Kev sib koom ua ke los ntawm cov khoom yog ib txoj kev tseem ceeb hauv kev xam lej uas tshwm sim ntau zaus hauv ntau qhov chaw, txij li physics thiab engineering mus rau economics thiab statistics. Hauv ntau qhov xwm txheej, cov integrals uas zoo li nyuaj lossis tsis daws tau siv cov txheej txheem txheem tuaj yeem ua kom yooj yim siv kev sib koom ua ke los ntawm cov khoom. Tsab xov xwm no yuav muab cov lus qhia ntxaws ntxaws txog yuav ua li cas daws cov integrals los ntawm cov khoom, suav nrog cov ntsiab lus yooj yim, cov kauj ruam daws teeb meem, thiab cov piv txwv ua tau.
Ib Feem Ntawm Kev Sib Koom Ua Ke Yog Dab Tsi?
Kev sib koom ua ke ib nrab yog ib txoj kev sib koom ua ke uas siv thaum ib qho kev sib koom ua ke yog cov khoom ntawm ob qho kev ua haujlwm uas yooj yim dua los ntawm kev faib nws ua ob ntu. Cov txheej txheem no yog raws li txoj cai ntawm kev sib koom ua ke ib nrab, uas yog kev siv txoj cai ntawm cov khoom sib txuas hauv kev suav lej sib txawv. Txoj cai yooj yim ntawm kev sib koom ua ke ib nrab tuaj yeem hais tau tias:
\[ \int u \, dv = uv – \int v \, du \]
Nov:
– \( u \) thiab \( v \) yog cov haujlwm uas yuav tsum tau txiav txim siab,
- \( du \) yog qhov derivative ntawm \( u \),
- \( dv \) yog qhov sib txawv ntawm \( v \), thiab
- \( dv \) raug koom ua ke kom tau txais \( v \).
Yuav siv txoj kev no, peb yuav tsum xaiv cov \(u\) thiab \(dv\) kom tsim nyog kom cov txheej txheem kev sib koom ua ke yooj yim dua tom qab siv cov mis.
Cov Kauj Ruam Los Daws Cov Kev Sib Piv Ib Nrab
1. Txheeb xyuas cov haujlwm \( u \) thiab \( dv \)
Kauj ruam thawj zaug hauv kev sib koom ua ke ib nrab yog xaiv cov functions \(u \) thiab \(dv \) rau qhov integral uas tau muab. Kev xaiv \(u \) thiab \(dv \) yog qhov tseem ceeb vim nws txiav txim siab qhov yooj yim ntawm qhov integral uas tau tshwm sim. Feem ntau, peb xaiv \(u \) ua lub function uas yooj yim dua thaum sib txawv (\(du \)), thiab \(dv \) ua lub function uas tseem yooj yim sib koom ua ke.
Ua ib qho kev qhia, peb siv tau txoj kev LIATE los xaiv \( u \ ):
- Cov haujlwm Logarithmic (\( \ln (x) \))
- Cov haujlwm trigonometric rov qab (\( \arctan (x), \arcsin (x), \arccos (x) \))
- Ib qho kev ua haujlwm lgebraic (x^n \))
- Cov kev ua haujlwm rigonometric (\( \sin(x), \cos(x) \))
- Cov haujlwm Exponential (\( e^x \))
Lub luag haujlwm uas tshwm sim thawj zaug hauv cov kab ke LIATE no feem ntau yog xaiv ua \( u \ ).
2. Muab \(u\) thiab sib xyaw \(dv\)
Tom qab xaiv \(u\) thiab \(dv\), kauj ruam tom ntej yog xam qhov derivative ntawm \(u\) (uas yog, \(du\)) thiab integral ntawm \(dv\) (uas yog, \(v\)).
3. Siv Cov Qauv Ib Nrab ntawm Kev Sib Koom Ua Ke
Thaum peb xam tau \(u\), \(du\), \(v\), thiab \(dv\), peb siv tau cov qauv integral ib nrab:
\[ \int u \, dv = uv – \int v \, du \]
4. Ua kom yooj yim thiab sib koom ua ke qhov seem Integral
Kauj ruam kawg yog ua kom yooj yim qhov tshwm sim thiab sib xyaw cov seem kom txog thaum peb tau txais qhov kev daws teeb meem kawg.
Piv txwv ua tau
Piv txwv 1: \( \int xe^x \, dx \)
Xav tias peb xav koom ua ke \( \int xe^x \, dx \).
1. Xaiv \( u \) thiab \( dv \):
- \( u = x \) (vim tias nws yooj yim dua thaum txo qis, dhau los ua 1)
– \( dv = e^x \, dx \) (vim nws yooj yim rau kev sib koom ua ke, nws tseem yog \( e^x \))
2. Tsim cov derivatives thiab integrals:
– \( du = dx \)
– \( v = \int e^x \, dx = e^x \)
3. Siv cov qauv integral ib nrab:
\[ \int xe^x \, dx = xe^x – \int e^x \, dx \]
4. Daws cov seem integral:
\[ \int e^x \, dx = e^x \]
Yog li ntawd:
\[ \int xe^x \, dx = xe^x – e^x + C \]
los yog
\[ \int xe^x \, dx = e^x (x – 1) + C \]
qhov twg \(C\) yog qhov tsis hloov pauv ntawm kev sib koom ua ke.
Piv txwv 2: \( \int \ln(x) \, dx \)
Xav tias peb xav koom ua ke \( \int \ln(x) \, dx \).
1. Xaiv \( u \) thiab \( dv \):
- \( u = \ln(x) \) (vim tias nws yooj yim dua thaum sib txawv)
– \( dv = dx \) (vim nws yooj yim rau kev sib koom ua ke)
2. Tsim cov derivatives thiab integrals:
– \( du = \frac{1}{x} \, dx \)
- \( v = \int dx = x\)
3. Siv cov qauv integral ib nrab:
\[ \int \ln(x) \, dx = x \ln(x) – \int x \left(\frac{1}{x} \, dx\right) \]
\[ \int \ln(x) \, dx = x \ln(x) – \int 1 \, dx \]
4. Daws cov seem integral:
\[ \int 1 \, dx = x \]
Yog li ntawd:
\[ \int \ln(x) \, dx = x \ln(x) – x + C \]
Cov Teeb Meem thiab Cov Lus Qhia
Cov Teeb Meem Feem Ntau
1. Xaiv tsis raug ntawm \(u \) thiab \(dv \): Xaiv tsis raug ntawm \(u \) thiab \(dv \) tuaj yeem ua rau qhov sib xyaw ua ke nyuaj dua. Kev ua raws li LIATE cov lus qhia feem ntau yuav pab tau.
2. Cov integrals uas tseem tshuav uas nyuaj: Qee zaum, tom qab siv cov mis integral ib nrab, cov integral uas tseem tshuav tseem nyuaj rau daws. Hauv qhov no, tej zaum yuav tsum tau siv cov txheej txheem integration ib nrab dua lossis siv lwm txoj kev.
Lub tswv yim
- Xyaum nrog ntau hom kev ua haujlwm sib txawv kom nkag siab txog cov qauv thiab txhim kho koj cov txuj ci hauv kev xaiv \(u\) thiab \(dv\).
- Siv kev sib xyaw ua ke ntawm cov txheej txheem kev sib koom ua ke yog tias tsim nyog, xws li u-substitution.
- Tsis txhob maj; xyuas txhua kauj ruam kom paub tseeb tias tsis muaj qhov yuam kev hauv kev txiav txim siab thiab kev koom ua ke.
Kev kaw
Cov kev suav lej ib nrab yog ib qho cuab yeej muaj zog hauv kev xam lej, uas yog ib txoj hauv kev yooj yim rau kev daws cov kev suav lej nyuaj. Los ntawm kev nkag siab txog cov ntsiab lus yooj yim thiab cov kauj ruam daws teeb meem kom raug, thiab xyaum nrog ntau yam piv txwv, peb tuaj yeem ua tus tswv ntawm cov txheej txheem no thiab siv nws rau ntau yam kev suav lej thiab kev tshawb fawb. Peb vam tias phau ntawv qhia no tau pab tau koj nkag siab thiab ntseeg siab daws cov kev suav lej ib nrab.
