Cov Lus Nug Piv Txwv Sib Tham Txog Cov Khoom ntawm Indexal Indexal
Tus lej indefinite integral yog ib lub tswv yim tseem ceeb hauv calculus, uas cuam tshuam nrog cov txheej txheem ntawm kev nrhiav qhov kev ua haujlwm qub los ntawm ib qho derivative. Cov txheej txheem no feem ntau hu ua antiderivative lossis kev koom ua ke. Ib qho tshwj xeeb ntawm tus lej indefinite integral yog tias qhov tshwm sim ntawm kev koom ua ke ib txwm suav nrog qhov tsis hloov pauv ntawm kev koom ua ke \ (C \) vim tias qhov sib txawv ntawm tus lej tas mus li yog xoom. Tsab xov xwm no yuav tham txog ntau qhov piv txwv ntawm cov lej indefinite integrals thiab tham txog cov khoom cuam tshuam nrog lawv.
1. Lub ntsiab lus ntawm Indefinite Integral
Tus lej sib xyaw tsis kawg ntawm ib qho kev ua haujlwm \(f(x) \) yog ib qho kev ua haujlwm \(F(x) \) uas nws cov derivative yog sib npaug rau \(f(x) \). Symbolically, yog tias \(F'(x) = f(x) \), ces:
\[
\int f(x) \, dx = F(x) + C
\]
qhov twg \(C\) yog qhov tsis hloov pauv ntawm kev sib koom ua ke.
2. Cov Khoom ntawm Indexal Indexal
Yuav kom yooj yim rau txoj kev sib koom ua ke, peb tuaj yeem siv ntau yam khoom dav dav ntawm cov indefinite integrals:
1. Cov Khoom Siv Linearity:
\[
\int [af(x) + bg(x)] \, dx = a \int f(x) \, dx + b \int g(x) \, dx
\]
qhov twg \(a\) thiab \(b\) yog cov constants.
2. Kev Sib Koom Ua Ib Ke ntawm Cov Tsis Hloov Pauv:
\[
\int k\, dx = kx + C
\]
qhov twg \(k\) yog ib qho tsis hloov pauv.
3. Kev Sib Koom Ua Ke ntawm Lub Hwj Chim:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]
rau \( n\neq -1\).
4. Kev Faib Tawm Ib Feem:
\[
\int (f(x) + g(x)) dx = \int f(x) \, dx + \int g(x) \, dx
\]
Los ntawm kev siv cov khoom no, peb tuaj yeem daws ntau yam teeb meem tsis kawg.
3. Piv txwv cov lus nug thiab kev sib tham
Piv txwv lus nug 1: Integral ntawm ib qho quadratic function
Lus Nug: Txheeb xyuas qhov sib xyaw ntawm \( f(x) = 3x^2 \).
Kev Sib Tham:
Peb siv cov khoom tseem ceeb ntawm lub hwj chim.
\[
\int 3x^2 \, dx
\]
\[
= 3 \int x^2 \, dx
\]
Los ntawm kev siv cov khoom sib xyaw ua ke:
\[
\int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^{3}}{3}
\]
Yog li ntawd:
\[
3 \int x^2 \, dx = 3 \cdot \frac{x^3}{3} = x^3
\]
Tsis txhob hnov qab ntxiv qhov kev sib koom ua ke tas li:
\[
\int 3x^2 \, dx = x^3 + C
\]
Piv txwv lus nug 2: Integrals ntawm trigonometric functions
Lus Nug: Txheeb xyuas qhov sib xyaw ntawm \( f(x) = \sin(x) \).
Kev Sib Tham:
Peb siv cov cuab yeej uas qhov sib xyaw ntawm \( \sin(x) \) yog \( -\cos(x) \):
\[
\int \sin(x) \, dx = -\cos(x) + C
\]
Yog li ntawd:
\[
\int \sin(x) \, dx = -\cos(x) + C
\]
Piv txwv 3: Integral ntawm ib qho exponential function
Lus Nug: Txheeb xyuas qhov sib xyaw ntawm \( f(x) = e^x \).
Kev Sib Tham:
Tus integral ntawm \( e^x \) tseem yog \( e^x \) vim tias cov khoom ntawm derivatives thiab exponential integrals zoo ib yam:
\[
\int e^x \, dx = e^x + C
\]
Piv txwv lus nug 4: Kev sib xyaw ua ke ntawm kev ua haujlwm sib xyaw
Lus Nug: Txheeb xyuas qhov sib xyaw ntawm \( f(x) = x^2 + 3x + 1 \).
Kev Sib Tham:
Peb siv tau cov khoom ntawm kev faib tawm integral:
\[
\int (x^2 + 3x + 1) \, dx = \int x^2 \, dx + \int 3x \, dx + \int 1 \, dx
\]
Siv cov khoom tseem ceeb ntawm txhua lub Cheebtsam:
\[
\int x^2 \, dx = \frac{x^3}{3}
\]
\[
\int 3x \, dx = 3 \int x \, dx = 3 \cdot \frac{x^2}{2} = \frac{3x^2}{2}
\]
\[
\int 1 \, dx = x
\]
Yog li ntawd:
\[
\int (x^2 + 3x + 1) \, dx = \frac{x^3}{3} + \frac{3x^2}{2} + x + C
\]
Piv txwv lus nug 5: Kev sib xyaw nrog kev hloov pauv yooj yim
Lus Nug: Txheeb xyuas qhov sib xyaw ntawm \( f(x) = (2x + 3)^5 \).
Kev Sib Tham:
Nov qhov kev hloov pauv \( u = 2x + 3 \) siv tau. Nrhiav qhov derivative \( du \):
\[
du = 2 \, dx \implies dx = \frac{1}{2} \, du
\]
Yog li ntawd, qhov sib npaug zos yuav yog:
\[
\int (2x + 3)^5 \, dx = \int u^5 \cdot \frac{dx}{du} \, du = \int u^5 \cdot \frac{1}{2} \, du = \frac{1}{2} \int u^5 \, du
\]
Kev koom ua ke \( u^5 \):
\[
\int u^5 \, du = \frac{u^6}{6}
\]
Yog li ntawd, qhov kawg yog:
\[
\frac{1}{2} \cdot \frac{u^6}{6} = \frac{u^6}{12}
\]
Hloov \( u \) nrog \( 2x + 3 \):
\[
\frac{(2x + 3)^6}{12} + C
\]
Piv txwv lus nug 6: Integral ntawm ib qho fractional function
Lus Nug: Txheeb xyuas qhov sib xyaw ntawm \( f(x) = \frac{1}{x} \).
Kev Sib Tham:
Peb paub tias qhov sib xyaw ua ke ntawm \( \frac{1}{x} \) yog \( \ln{|x|} \):
\[
\int \frac{1}{x} \, dx = \ln{|x|} + C
\]
4. Txoj kev
Tus lej indefinite integral yog ib qho cuab yeej tseem ceeb heev hauv kev xam lej rau kev nrhiav qhov function thawj los ntawm ib qho derivative uas paub lawm. Cov yam ntxwv ntawm linearity, tus lej integral ntawm ib qho constant, qhov distributivity ntawm integrals, thiab lwm yam yog qhov pab tau zoo heev rau hauv cov txheej txheem kev sib koom ua ke. Yog tias muaj kev xyaum txaus, ntau hom integrals tuaj yeem daws tau zoo.
Los ntawm kev nkag siab txog cov ntsiab lus tseem ceeb thiab cov yam ntxwv ntawm cov indefinite integrals, vam tias cov tub ntxhais kawm yuav pom tias nws yooj yim dua los daws ntau yam teeb meem uas cuam tshuam nrog cov indefinite integrals. Kev xyaum txuas ntxiv mus yuav ua rau lawv nkag siab thiab muaj peev xwm siv cov indefinite integrals hauv ntau yam lej.
