Me pēhea te whakaoti i ngā taunga whakauru ā-wāhanga: He aratohu katoa
He tikanga nui te whakaurunga-ā-wāhanga i roto i te tātaitai e puta pinepine ana i roto i ngā momo mara, mai i te ahupūngao me te miihini ki te ōhanga me te tatauranga. I roto i te nuinga o ngā wā, ko ngā whakaurunga e ahua uaua ana, e kore rānei e taea te whakaoti mā te whakamahi i ngā tikanga paerewa ka taea te whakangawari mā te whakamahi i te whakaurunga-ā-wāhanga. Ka whakaratohia e tēnei tuhinga he aratohu taipitopito mō te whakaoti rapanga whakauru-ā-wāhanga, tae atu ki ngā ariā taketake, ngā mahi whakaoti, me ngā tauira mahi.
He aha te Taurite Wāhanga?
Ko te whakaurunga ā-wāhanga he tikanga whakaurunga e whakamahia ana ina he hua o ngā mahi e rua te whakaurunga, ā, he māmā ake te whakahaere mā te wehewehe kia rua ngā wāhanga. E hangai ana tēnei tikanga ki te ture o te whakaurunga ā-wāhanga, arā, he whakamahinga o te ture hua-pānga i roto i te tātaitai rerekētanga. Ka taea te kī ko te ture taketake o te whakaurunga ā-wāhanga penei:
\[ \int u \, dv = uv – \int v \, du \]
Anei:
– Ko ngā mahi a \( u \) me \( v \) hei whakatau,
– Ko te \( du \) te pūrua o \( u \),
– Ko te \( dv \) te rerekētanga o \( v \), me
– Kua whakauruhia a \( dv \) hei whiwhi i te \( v \).
Hei whakamahi i tēnei tikanga, me whiriwhiri tātou i te \( u \) me te \( dv \) e tika ana kia māmā ake ai te tukanga whakauru i muri i te whakamahinga o te tātai.
Ngā Hipanga hei Whakatau i ngā Taunga Whakauru Wāhanga
1. Tāutuhia ngā mahi \( u \) me \( dv \)
Ko te taahiraa tuatahi i roto i te whakaurunga ā-wāhanga ko te whiriwhiri i ngā mahi \( u \) me \( dv \) mō te whakaurunga kua hoatu. He mea nui te whiriwhiri i te \( u \) me te \( dv \) nā te mea ka whakatau i te māmā o te whakaurunga ka puta. I te nuinga o te wā, ka whiriwhiria e tātou te \( u \) hei mahi ka māmā ake i te wehewehenga (\( du \)), me te \( dv \) hei mahi ka noho ngāwari ki te whakauru.
Hei aratohu, ka taea e tātou te whakamahi i te tikanga LIATE hei tīpako i te \( u \):
– Ngā mahi taupū (\( \ln (x) \))
– Ngā mahi pārōnaki whakamuri (\( \arctan(x), \arcsin(x), \arccos(x) \))
– Ngā mahi ā-tauira (\( x^n \))
– Ngā mahi ā-toru-ira (\( \sin(x), \cos(x) \))
– Ngā mahi taupū (\( e^x \))
Ko te mahi e puta tuatahi mai ana i tēnei raupapa LIATE ka whiriwhiria i te nuinga o te wā ko \( u \).
2. Tangohia \( u \) me te Whakauru \( dv \)
I muri i te whiriwhiri i a \( u \) me \( dv \), ko te mahi e whai ake nei ko te tatau i te tauwehenga o \( u \) (arā, \( du \)) me te tauwehenga o \( dv \) (arā, \( v \)).
3. Whakamahia te Tātai Whakakotahi Wāhanga
Kia oti te tatau i a \( u \), \( du \), \( v \), me \( dv \), ka taea e tātou te whakamahi i te tātai tauwehe wāhanga:
\[ \int u \, dv = uv – \int v \, du \]
4. Whakangāwaritia, ā, Whakaurua te Taurite e Toe Ana
Ko te taahiraa whakamutunga ko te whakangawari i te hua me te whakauru i te toenga kia whiwhi ra ano i te otinga whakamutunga.
Ngā Tauira Whaihua
Tauira 1: \( \int xe^x \, dx \)
Me kī tātou e hiahia ana ki te whakauru i te \( \int xe^x \, dx \).
1. Tīpakohia \( u \) me \( dv \):
– \( u = x \) (nā te mea ka māmā ake ina whakaitihia, ka noho hei 1)
– \( dv = e^x \, dx \) (nā te mea he māmā te whakauru, ka noho tonu \( e^x \))
2. Whakaputahia ngā taupū me ngā taupū whakauru:
– \( du = dx \)
– \( v = \int e^x \, dx = e^x \)
3. Whakamahia te tātai tauwehenga ā-wāhanga:
\[ \int xe^x \, dx = xe^x – \int e^x \, dx \]
4. Whakaotia te tauwehenga toenga:
\[ \int e^x \, dx = e^x \]
Nā reira:
\[ \int xe^x \, dx = xe^x – e^x + C \]
ranei
\[ \int xe^x \, dx = e^x (x – 1) + C \]
ko \( C \) te pūmau o te whakaurunga.
Tauira 2: \( \int \ln(x) \, dx \)
Me kī tātou e hiahia ana ki te whakauru i te \( \int \ln(x) \, dx \).
1. Tīpakohia \( u \) me \( dv \):
– \( u = \ln(x) \) (nā te mea ka māmā ake ina wehewehea)
– \( dv = dx \) (nā te mea he māmā te whakauru)
2. Whakaputahia ngā taupū me ngā taupū whakauru:
– \( du = \frac{1}{x} \, dx \)
– \( v = \int dx = x \)
3. Whakamahia te tātai tauwehenga ā-wāhanga:
\[ \int \ln(x) \, dx = x \ln(x) – \int x \left(\frac{1}{x} \, dx\matau) \]
\[ \int \ln(x) \, dx = x \ln(x) – \int 1 \, dx \]
4. Whakaotia te tauwehenga toenga:
\[ \int 1 \, dx = x \]
Nā reira:
\[ \int \ln(x) \, dx = x \ln(x) – x + C \]
Ngā Uauatanga me ngā Tohutohu
Ngā Uauatanga Noa
1. Kōwhiringa hē o \( u \) me \( dv \): Mā te kōwhiringa hē o \( u \) me \( dv \) ka uaua ake te whakauru. He pai te whai i te aratohu a LIATE.
2. Ngā taunga whakauru uaua e toe ana: I ētahi wā, i muri i te whakamahi i te tātai taunga whakauru wāhanga, he uaua tonu te whakaoti i te taunga whakauru e toe ana. I tēnei wā, me whakamahi anō pea i te tikanga whakauru wāhanga, me whakamahi rānei i tētahi atu tikanga.
Tips
– Whakaharatauhia ngā momo mahi rerekē kia mārama ai ki ngā tauira, kia whakakoi ai hoki i ō pūkenga ki te whiriwhiri i te \( u \) me te \( dv \).
– Whakamahia he huinga o ngā tikanga whakauru mēnā e tika ana, pērā i te whakakapinga-u.
– Kaua e hohoro; tirohia ia taahiraa kia kore ai he hapa i roto i te whakaputa me te whakauru.
Te Katinga
He taputapu kaha ngā taunga ā-wāhanga i roto i te tātaitai, e whakatakoto ana i te ara mō te whakaoti rapanga taunga uaua i roto i te huarahi māmā ake. Mā te mārama ki ngā ariā taketake me ngā mahi whakaoti tika, me te whakaharatau ki ngā tauira rerekē, ka taea e tātou te matatau ki tēnei tikanga me te whakamahi i roto i ngā horopaki pāngarau me te pūtaiao. Ko te tumanako kua whai hua tēnei aratohu hei āwhina i a koe ki te mārama me te whakaoti rapanga taunga ā-wāhanga.
