Piv txwv ntawm Cov Lus Nug Sib Tham Txog Kev Sib Tham Tsis Paub Kawg
Tus lej indefinite integral yog ib lub tswv yim tseem ceeb hauv calculus, siv los nrhiav tus function thawj los ntawm tus function uas tau muab los. Tus lej indefinite integral yog cim los ntawm lub cim ∫ ua raws li tus function uas yuav tsum tau integrate thiab tus variable ntawm kev integration. Hauv tsab xov xwm no, peb yuav tham txog ntau yam piv txwv ntawm cov lej indefinite integrals thiab lawv cov kev daws teeb meem.
Piv txwv lus nug 1: Kev sib xyaw ua ke ntawm Polynomial Functions
Lus Nug: Txheeb xyuas qhov sib xyaw ntawm qhov kev ua haujlwm \( f(x) = 3x^2 \).
Kev Sib Tham: Txhawm rau koom ua ke cov haujlwm polynomial, peb siv cov cai yooj yim ntawm kev koom ua ke, uas yog:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Siv cov cai no, qhov sib xyaw ntawm \(3x^2\) yog:
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) + C = 3 \left( \frac{1}{3} x^3 \right) + C = x^3 + C \]
Yog li, \( \int 3x^2 \, dx = x^3 + C \).
Piv txwv lus nug 2: Kev sib xyaw ua ke ntawm Exponential Functions
Lus Nug: Txheeb xyuas qhov sib xyaw ua ke ntawm qhov kev ua haujlwm \( f(x) = e^x \).
Kev Sib Tham: Qhov integral ntawm exponential function \( e^x \) yooj yim heev vim tias qhov function \( e^x \) yog ib qho function uas tsis hloov pauv kiag li nyob rau hauv ob qho kev ua haujlwm sib txawv thiab integral:
\[ \int e^x \, dx = e^x + C \]
Yog li, \( \int e^x \, dx = e^x + C \).
Piv txwv lus nug 3: Kev sib xyaw ua ke ntawm Trigonometric Functions
Lus Nug: Txheeb xyuas qhov sib xyaw ntawm qhov kev ua haujlwm \( f(x) = \sin(x) \).
Kev Sib Tham: Yuav kom koom ua ke cov haujlwm trigonometric, peb yuav tsum paub cov integrals yooj yim ntawm cov haujlwm ntawd. Ib qho ntawm cov kev sib raug zoo yooj yim yog:
\[ \int \sin(x) \, dx = -\cos(x) + C \]
Yog li, \( \int \sin(x) \, dx = -\cos(x) + C \).
Piv txwv lus nug 4: Kev sib xyaw ua ke ntawm cov haujlwm fractional
Lus Nug: Txheeb xyuas qhov sib xyaw ntawm qhov kev ua haujlwm \( f(x) = \frac{1}{x} \).
Kev Sib Tham: Qhov sib xyaw ua ke ntawm qhov kev ua haujlwm \( \frac{1}{x} \) yog:
\[ \int \frac{1}{x} \, dx = \ln|x| +C\]
Yog li, \( \int \frac{1}{x} \, dx = \ln|x| + C \).
Piv txwv lus nug 5: Kev sib xyaw ua ke ntawm cov haujlwm tsis zoo Exponential
Lus Nug: Txheeb xyuas qhov sib xyaw ntawm qhov kev ua haujlwm \( f(x) = x^{-2} \).
Kev Sib Tham: Rau \( n \neq -1 \), peb siv txoj cai integral yooj yim:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Hauv qhov xwm txheej no, \( n = -2 \), yog li:
\[ \int x^{-2} \, dx = \int x^{-2} \, dx = \frac{1}{-2+1} x^{-2+1} + C = \frac{1}{-1} x^{-1} + C = -x^{-1} + C = -\frac{1}{x} + C \]
Yog li, \( \int x^{-2} \, dx = -\frac{1}{x} + C \).
Piv txwv lus nug 6: Kev sib xyaw ua ke ntawm cov haujlwm sib xyaw ua ke
Lus Nug: Txheeb xyuas qhov sib xyaw ua ke ntawm qhov kev ua haujlwm \( f(x) = 4x^3 – 3x^2 + 2x – 5 \).
Kev Sib Tham: Peb tuaj yeem sib koom ua ke txhua lo lus sib cais siv cov cai yooj yim ntawm kev sib koom ua ke:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = \int 4x^3 \, dx – \int 3x^2 \, dx + \int 2x \, dx – \int 5 \, dx \]
Tam sim no peb muab txhua lo lus sib txuas ua ke:
\[ \int 4x^3 \, dx = 4 \int x^3 \, dx = 4 \left( \frac{1}{3+1} x^{3+1} \right) = 4 \left( \frac{1}{4} x^4 \right) = x^4 \]
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) = 3 \left( \frac{1}{3} x^3 \right) = x^3 \]
\[ \int 2x \, dx = 2 \int x \, dx = 2 \left( \frac{1}{1+1} x^{1+1} \right) = 2 \left( \frac{1}{2} x^2 \right) = x^2 \]
\[ \int 5 \, dx = 5x \]
Ua ke cov txiaj ntsig no, peb tau txais:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \]
Yog li, \( \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \).
Xaus
Tus lej indefinite integral yog ib lub tswv yim tseem ceeb heev hauv calculus thiab muaj ntau txoj cai uas ua rau nws yooj yim dua los koom ua ke ntau hom kev ua haujlwm. Hauv tsab xov xwm no, peb tau tham txog ntau qhov piv txwv ntawm cov lej indefinite integrals, suav nrog polynomials, exponentials, trigonometric functions, fractions, functions nrog negative exponents, thiab kev sib xyaw ua ke ntawm cov functions. Kev nkag siab thiab kev ua tus tswv ntawm cov cai yooj yim ntawm cov lej integrals yuav pab tau zoo heev rau kev daws ntau yam teeb meem calculus.
Cov integrals tsis kawg tsis yog tsuas yog tseem ceeb hauv kev xav lej xwb, tab sis kuj muaj kev siv dav hauv physics, engineering, thiab lwm yam teb. Yog tias xyaum txaus, kev koom ua ke ntawm ntau yam functions yuav yooj yim dua thiab nkag siab dua.
