Tauira o ngā Pātai Kōrero Whakauru Kore-Mutu
Ko te taupū mutunga kore he ariā taketake i roto i te tātaitai, e whakamahia ana hei kimi i te mahi taketake mai i tētahi mahi i ahu mai. Ko te taupū mutunga kore e tohuhia e te tohu ∫, ā, muri iho ko te mahi hei whakauru me te taurangi whakauru. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira o ngā taupū mutunga kore me ā rātou otinga.
Tauira Pātai 1: Te Whakapūtanga o ngā Mahi Pūrau
Pātai: Whakatauhia te tauwehenga o te mahi \( f(x) = 3x^2 \).
Kōrero: Hei whakauru i ngā mahi pūrau, ka whakamahia e mātou ngā ture taketake o te whakauru, arā:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Mā te whakamahi i ēnei ture, ko te taupū o \( 3x^2 \) ko:
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) + C = 3 \left( \frac{1}{3} x^3 \right) + C = x^3 + C \]
Nō reira, \( \int 3x^2 \, dx = x^3 + C \).
Tauira Pātai 2: Te Whakapūtanga o ngā Mahi Taupūtanga
Pātai: Whakatauhia te tauwehenga o te mahi \( f(x) = e^x \).
Kōrero: He tino māmā te taupū o te mahi taupūpū \( e^x \) nā te mea ko te mahi \( e^x \) he mahi e kore e rerekē i raro i ngā mahi rerekētanga me ngā mahi taupūpū:
\[ \int e^x \, dx = e^x + C \]
Nō reira, \( \int e^x \, dx = e^x + C \).
Tauira Pātai 3: Te Whakapūtanga o ngā Mahi Pāngatoru
Pātai: Whakatauhia te tauwehenga o te mahi \( f(x) = \sin(x) \).
Kōrero: Hei whakauru i ngā mahi taurite, me mōhio tātou ki ngā taurite taketake o aua mahi. Ko tētahi o ngā whanaungatanga taketake ko:
\[ \int \sin(x) \, dx = -\cos(x) + C \]
Nō reira, \( \int \sin(x) \, dx = -\cos(x) + C \).
Tauira Pātai 4: Te Whakapūtanga o ngā Taumahi Hautau
Pātai: Whakatauhia te tauwehenga o te mahi \( f(x) = \frac{1}{x} \).
Kōrero: Ko te taupū o te mahi \( \frac{1}{x} \) ko:
\[ \int \frac{1}{x} \, dx = \ln|x| +C\]
Nō reira, \( \int \frac{1}{x} \, dx = \ln|x| + C \).
Tauira Pātai 5: Te Whakapūtanga o ngā Mahi Taupūnga Kino
Pātai: Whakatauhia te tauwehenga o te mahi \( f(x) = x^{-2} \).
Kōrero: Mō \( n \neq -1 \), ka whakamahia e mātou te ture taupū taketake:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
I tēnei wā, \( n = -2 \), nō reira:
\[ \int x^{-2} \, dx = \int x^{-2} \, dx = \frac{1}{-2+1} x^{-2+1} + C = \frac{1}{-1} x^{-1} + C = -x^{-1} + C = -\frac{1}{x} + C \]
Nō reira, \( \int x^{-2} \, dx = -\frac{1}{x} + C \).
Tauira Pātai 6: Te Whakapūtanga o ngā Mahi Whakakotahi
Pātai: Whakatauhia te tauwehenga o te mahi \( f(x) = 4x^3 – 3x^2 + 2x – 5 \).
Kōrero: Ka taea e tātou te whakauru motuhake i ia kupu mā te whakamahi i ngā ture taketake o te whakauru:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = \int 4x^3 \, dx – \int 3x^2 \, dx + \int 2x \, dx – \int 5 \, dx \]
Inaianei ka whakauruhia e mātou ia kupu takitahi:
\[ \int 4x^3 \, dx = 4 \int x^3 \, dx = 4 \left( \frac{1}{3+1} x^{3+1} \right) = 4 \left( \frac{1}{4} x^4 \right) = x^4 \]
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \left( \frac{1}{2+1} x^{2+1} \right) = 3 \left( \frac{1}{3} x^3 \right) = x^3 \]
\[ \int 2x \, dx = 2 \int x \, dx = 2 \left( \frac{1}{1+1} x^{1+1} \right) = 2 \left( \frac{1}{2} x^2 \right) = x^2 \]
\[ \int 5 \, dx = 5x \]
Mā te whakakotahi i ēnei hua, ka whiwhi tātou:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \]
Nō reira, \( \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \).
Whakamutunga
He ariā tino nui te taupū mutunga kore i roto i te tātaitai, ā, he maha ngā ture e āwhina ana i te whakauru i ngā momo mahi rerekē. I roto i tēnei tuhinga, kua matapakihia e mātou ētahi tauira o ngā taupū mutunga kore, tae atu ki ngā pūrinomiara, ngā taupūtanga, ngā mahi whārite, ngā hautau, ngā mahi me ngā taupūtanga kino, me ngā huinga mahi. Mā te mārama me te matatau ki ēnei ture taketake o ngā taupūtanga ka tino āwhina i te whakaoti rapanga tātaitai.
Ehara i te mea he mea nui ngā taunga taurite kore mutunga i roto i te ariā pāngarau anake, engari he whānui hoki ngā tono i roto i te ahupūngao, te hangarau, me ētahi atu mara. Mēnā ka nui te mahi, ka māmā ake, ka mārama ake hoki te whakauru i ngā mahi maha.
