Tauira Pātai me te Kōrero: Whakamāramatanga o te Taupū Kore-Mutu
Ko te taupū mutunga kore he ariā taketake i roto i te tātaitai, e whakamahia ana hei kimi i te taupū ā-whakakore o tētahi mahi kua hoatu. E mōhiotia ana hoki ko te ā-whakarerekētanga. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira o ngā taupū mutunga kore, me ngā whakamārama, kia pai ake ai te mārama ki tēnei ariā.
Te Mārama ki ngā Whakaurunga Kore-Mutu
Ko te taupū mutunga kore te tukanga kimi i te mahi taketake \( F(x) \) mai i tētahi taupū \( f(x) \), e tohuhia ana e:
\[ \int f(x) \, dx = F(x) + C \]
ko \( C \) he pūmau whakauru. Ka puta tēnei pūmau nā te mea he kore te pānga o tētahi pūmau, nō reira i roto i te tukanga ārai-whakarerekētanga, me whakaaro tātou ki te āheinga o tētahi pūmau pēnei.
Te Tātai Taketake mō ngā Taurite Kore-Mutu
Ko ētahi o ngā tātai taketake e whakamahia ana i roto i ngā taupū mutunga kore ko:
1. \[ \int k \, dx = kx + C \]
ina he pūmau a \( k \).
2. \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
mō \( n \neq -1 \).
3. \[ \int e^x \, dx = e^x + C \]
4. \[ \int a^x \, dx = \frac{a^x}{\ln a} + C \]
ko \( a \) he tau tūturu pai, ā, ko \( a \neq 1 \).
5. \[ \int \frac{1}{x} \, dx = \ln |x| +C\]
6. \[ \int \sin x \, dx = -\cos x + C \]
7. \[ \int \cos x \, dx = \sin x + C \]
Ngā Tauira o ngā Raru Whakauru Kore-Mutu me ā Rātou Kōrero
Tauira 1
Pātai:
Tātaihia te taupū mutunga kore o \( f(x) = 3x^2 \).
Kōrero:
Hei whakaoti i tēnei taupū, ka whakamahia e mātou te tātai taupū taketake mō ngā mahi o te āhua \( x^n \):
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
I tēnei wā, kei a tātou \( f(x) = 3x^2 \), ko \( k = 3 \) me \( n = 2 \). Kātahi:
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{x^{3}}{3} \right) + C = x^3 + C \]
Nō reira, \( \int 3x^2 \, dx = x^3 + C \).
Tauira 2
Pātai:
Tātaihia te taupū mutunga kore o \( f(x) = \frac{1}{x} \).
Kōrero:
Ko te taupū o \( \frac{1}{x} \) i runga i te tātai taketake ko:
\[ \int \frac{1}{x} \, dx = \ln |x| +C\]
Nō reira, \( \int \frac{1}{x} \, dx = \ln |x| + C \).
Tauira 3
Pātai:
Tātaihia te taupū mutunga kore o \( f(x) = e^x \).
Kōrero:
Ko te taupū o \( e^x \) i runga i te tātai taketake ko:
\[ \int e^x \, dx = e^x + C \]
Nō reira, \( \int e^x \, dx = e^x + C \).
Tauira 4
Pātai:
Tātaihia te taupū mutunga kore o \( \sin x \).
Kōrero:
Ko te taupū o \( \sin x \) i runga i te tātai taketake ko:
\[ \int \sin x \, dx = -\cos x + C \]
Nō reira, \( \int \sin x \, dx = -\cos x + C \).
Tauira 5
Pātai:
Tātaihia te taupū mutunga kore o \( \cos x \).
Kōrero:
Ko te taupū o \( \cos x \) i runga i te tātai taketake ko:
\[ \int \cos x \, dx = \sin x + C \]
Nō reira, \( \int \cos x \, dx = \sin x + C \).
Tauira 6
Pātai:
Tātaihia te taupū mutunga kore o \( 5x^{-3} \).
Kōrero:
Hei whakaoti i tēnei taupū, ka whakamahia e mātou te tātai taupū taketake mō ngā mahi o te āhua \( x^n \):
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
I tēnei wā, kei a tātou \( f(x) = 5x^{-3} \), ko \( k = 5 \) me \( n = -3 \). Kātahi:
\[ \int 5x^{-3} \, dx = 5 \int x^{-3} \, dx = 5 \left( \frac{x^{-3+1}}{-3+1} \right) + C = 5 \left( \frac{x^{-2}}{-2} \right) + C = -\frac{5}{2} x^{-2} + C \]
Nō reira, \( \int 5x^{-3} \, dx = -\frac{5}{2} x^{-2} + C \).
Tauira 7
Pātai:
Tātaihia te taupū mutunga kore o \( 4e^{2x} \).
Kōrero:
Hei whakaoti i tēnei taupū, me whakamahi tātou i te matū \( u \). Me whakatakoto e tātou \( u = 2x \) kia \( du = 2dx \), kia \( dx = \frac{du}{2} \ rānei).
\[ \int 4e^{2x} \, dx = 4 \int e^{u} \, \frac{du}{2} = 2 \int e^{u} \, du \]
Nā, ko te taupū o \( e^u \) ko \( e^u \):
\[ 2 \int e^u \, du = 2e^u + C \]
Hoki ki ngā taurangi taketake:
\[ 2e^u + C = 2e^{2x} + C \]
Nō reira, \( \int 4e^{2x} \, dx = 2e^{2x} + C \).
Te Whakamahinga o ngā Taurite Kore-Mutu
He whānuitia ngā whakamahinga o ngā taunga mutunga kore i roto i te pūtaiao me te miihini. Hei tauira, i roto i te ahupūngao, ka whakamahia ngā taunga mutunga kore hei kimi i te tawhiti i haerea e tētahi mea ina mōhiotia tōna tere hei pānga o te wā. I roto i te ōhanga, ka taea te whakamahi i ngā taunga mutunga kore hei kimi i te utu katoa, i te hua rānei ina mōhiotia te tere o te huringa o te utu, o te hua rānei mō ia wae.
Whakamutunga
He ariā nui te taupū mutunga kore i roto i te tātaitai, e whakamahia ana hei kimi i ngā taupatupatu ārai-ira o ngā mahi. He mea nui te mārama ki ngā tātai taupū taketake hei whakaoti rapanga e pā ana ki ngā taupū mutunga kore. Mā te whakamahi i ngā tauira pēnei i ngā mea kua kōrerohia i runga ake nei, ka taea e te tangata te mōhio ki te tikanga o ngā taupū mutunga kore. Ehara i te mea he uara ariā anake te ariā o ngā taupū mutunga kore engari he uara mahi anō hoki i roto i ngā momo mara pūtaiao.
