Piv txwv ntawm Cov Lus Nug Sib Tham Ua Ke
Kev suav lej (Integral) yog ib lub tswv yim tseem ceeb hauv kev suav lej uas muaj kev siv dav hauv ntau qhov chaw, suav nrog physics, engineering, thiab economics. Tsab xov xwm no yuav tshawb nrhiav ntau yam piv txwv ntawm cov teeb meem suav lej thiab lawv cov kev daws teeb meem kom nkag siab tob dua.
1. Kev Nkag Siab Txog Cov Integrals
Hauv cov lus yooj yim, ib qho integral yog qhov kev ua haujlwm inverse ntawm ib qho derivative. Muaj ob hom integrals uas feem ntau tham txog, uas yog:
- Indefinite Integral: qhov no yog daim ntawv integral uas tsis muaj kev txwv sab saud thiab sab qis thiab yog cim los ntawm ∫ f(x) dx.
- Kev Sib Koom Ua Ib Ke: qhov no yog daim ntawv sib koom ua ke uas muaj cov kev txwv sab saud thiab sab qis thiab yog cim los ntawm ∫[a,b] f(x) dx.
Tus indefinite integral feem ntau hu ua anti-derivative, thiab qhov tshwm sim yuav suav nrog tus C tas mus li vim tias cov khoom ntawm tus derivative tas mus li yog xoom.
2. Piv txwv ntawm Cov Teeb Meem Indexal Indexal
Piv txwv 1: Kev Sib Piv Yooj Yim Tsis Paub Txwm
Xam ∫ x^2 dx.
Kev Sib Tham:
Peb paub tias txoj cai sib koom ua ke yooj yim rau ∫ x^n dx yog (x^(n+1))/(n+1) + C, qhov twg C yog qhov tsis hloov pauv ntawm kev sib koom ua ke.
Rau qhov kev sib xyaw saum toj no, n = 2:
∫ x^2 dx = (x^(2+1))/(2+1) + C
= (x^3)/3 + C.
Yog li, qhov tshwm sim ntawm ∫ x^2 dx yog (x^3)/3 + C.
Piv txwv 2: Kev sib xyaw ua ke ntawm Exponential Functions
Xam ∫ e^x dx.
Kev Sib Tham:
Txoj cai yooj yim rau qhov exponential integral ∫ e^x dx yog e^x + C.
Yog li, qhov tshwm sim ntawm ∫ e^x dx yog e^x + C.
3. Piv txwv ntawm Cov Teeb Meem Sib Xws
Piv txwv 1: Kev Sib Piv Yooj Yim
Xam ∫[1,3] x^2 dx.
Kev Sib Tham:
Ua ntej, peb nrhiav tau qhov anti-derivative ntawm x^2, uas yog (x^3)/3.
Tam sim no peb hloov cov kev txwv:
∫[1,3] x^2 dx = [(3^3)/3 – (1^3)/3]
= [27/3 – 1/3]
= [9 – 1/3]
= 8 + 2/3 los yog 8.6667.
Yog li, qhov tshwm sim ntawm ∫[1,3] x^2 dx yog 26/3 lossis 8.6667.
Piv txwv 2: Kev sib koom ua ke los ntawm kev hloov pauv
Xam ∫[0,2] (2x + 1) dx.
Kev Sib Tham:
Ua ntej, peb nrhiav tau qhov antiderivative ntawm 2x + 1, uas yog x^2 + x. Tam sim no peb hloov cov kev txwv:
∫[0,2] (2x+1) dx = [(2^2 + 2) – (0^2 + 0)]
= [(4 + 2) – 0]
= 6.
Yog li, qhov tshwm sim ntawm ∫[0,2] (2x + 1) dx yog 6.
4. Piv txwv ntawm Cov Teeb Meem Integral nrog Partial Method
Ib feem ntawm kev sib xyaw ua ke yog ib txoj kev siv thaum qhov kev sib xyaw ua ke ntawm cov khoom ntawm ob qho kev ua haujlwm nyuaj rau xam ncaj qha. Cov mis rau kev sib xyaw ua ke ib nrab yog:
∫ u dv = uv – ∫ v du
Piv txwv: Trigonometric Partial Integrals
Xam ∫ xe^x dx.
Kev Sib Tham:
Nov peb siv txoj kev faib ua feem. Xav tias u = x thiab dv = e^x dx. Ces du = dx thiab v = e^x.
Raws li cov qauv sib xyaw ua ke ib nrab:
∫ xe^x dx = xe^x – ∫ e^x dx
= xe^x – e^x + C
= e^x(x – 1) + C.
Yog li, qhov tshwm sim ntawm ∫ xe^x dx yog e^x(x – 1) + C.
5. Piv txwv ntawm Cov Teeb Meem Trigonometric Integral
Piv txwv: Kev Sib Koom Ua Ke ntawm Cov Haujlwm Trigonometric Yooj Yim
Xam ∫ cos(x) dx.
Kev Sib Tham:
Txoj cai yooj yim ntawm kev koom ua ke ntawm cos (x) yog sin (x) + C.
Yog li, qhov tshwm sim ntawm ∫ cos(x) dx yog sin(x) + C.
Piv txwv: Kev sib xyaw ua ke ntawm Trigonometric Functions nrog rau kev txwv
Xam ∫[0,π/2] sin(x) dx.
Kev Sib Tham:
Ua ntej, peb pom qhov anti-derivative ntawm sin(x), uas yog -cos(x).
Tam sim no, hloov cov kev txwv:
∫[0,π/2] sin(x) dx = [ -cos(π/2) – (-cos(0))]
= [ -0 – (-1) ]
= 1.
Yog li, qhov tshwm sim ntawm ∫[0,π/2] sin(x) dx yog 1.
6. Piv txwv ntawm Kev Hloov Chaw Ua Ib Feem
Piv txwv: Kev Hloov Chaw Sib Koom Tes
Xam ∫ 2x sqrt(1-x^2) dx.
Kev Sib Tham:
Siv qhov kev hloov pauv u = 1-x^2, ces du = -2x dx.
Tom qab ntawd qhov kev hloov pauv tseem ceeb mus rau:
∫ 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.
Yog li, qhov tshwm sim ntawm ∫ 2x sqrt(1-x^2) dx yog -1/3 (1-x^2)^(3/2) + C.
7. Txoj kev
Cov integrals yog ib qho cuab yeej siv tau zoo heev hauv kev lej rau kev nrhiav thaj tsam hauv qab ib daim duab nkhaus, ntim, thiab ntau lwm yam kev siv. Kev nkag siab txog ntau yam kev sib koom ua ke, xws li kev hloov pauv, partials, thiab cov hauv paus ntawm integrals, yog qhov tseem ceeb. Cov piv txwv uas tau tham saum toj no vam tias yuav pab txhawb koj txoj kev nkag siab txog lub tswv yim ntawm integrals.
Kev xyaum ua tas li thiab kev nkag siab txog lub ntsiab lus yog qhov tseem ceeb rau kev paub txog kev sib xyaw ua ke. Txuas ntxiv xyaum nrog ntau yam hloov pauv thiab ntau yam kev ua haujlwm kom koj paub ntau ntxiv txog qhov no.
Vam tias tsab xov xwm no yuav pab tau koj hauv kev kawm txog integrals.
