Momwe Mungathetsere Zophatikiza Zapadera: Buku Lathunthu
Kuphatikiza ndi zigawo ndi njira yofunika kwambiri mu calculus yomwe imapezeka kawirikawiri m'magawo osiyanasiyana, kuyambira pa fizikisi ndi uinjiniya mpaka zachuma ndi ziwerengero. Nthawi zambiri, zinthu zophatikiza zomwe zimawoneka zovuta kapena zosatheka kuthetsedwa pogwiritsa ntchito njira zokhazikika zitha kuphweka pogwiritsa ntchito kuphatikiza ndi zigawo. Nkhaniyi ipereka chitsogozo chatsatanetsatane cha momwe mungathetsere zinthu zophatikiza ndi zigawo, kuphatikiza mfundo zoyambira, njira zothetsera mavuto, ndi zitsanzo zothandiza.
Kodi Partial Integral ndi chiyani?
Kuphatikizana kwa magawo ndi njira yolumikizirana yomwe imagwiritsidwa ntchito pamene chinthu chofunikira ndi chinthu chopangidwa ndi ntchito ziwiri zomwe zimagwiridwa mosavuta pochigawa m'magawo awiri. Njirayi imachokera pa lamulo la kuphatikizana kwa magawo, lomwe ndi kugwiritsa ntchito lamulo la zotumphukira-zogulitsa mu kuwerengera kosiyana. Lamulo loyambira la kuphatikizana kwa magawo lingatchulidwe motere:
\[ \int u \, dv = uv – \int v \, du \]
Pano:
– \( u \) ndi \( v \) ndi ntchito zomwe ziyenera kudziwika,
– \( du \) ndi chochokera ku \( u \),
– \( dv \) ndiye kusiyana kwa \( v \), ndipo
– \( dv \) yaphatikizidwa kuti ipeze \( v \).
Kuti tigwiritse ntchito njira iyi, tifunika kusankha \( u \) ndi \( dv \) zoyenera kuti njira yolumikizira ikhale yosavuta titagwiritsa ntchito fomula.
Njira Zothetsera Zophatikiza Zochepa
1. Dziwani ntchito \( u \) ndi \( dv \)
Gawo loyamba mu kuphatikiza pang'ono ndikusankha ntchito \( u \) ndi \( dv \) za integral yomwe yaperekedwa. Kusankha \( u \) ndi \( dv \) ndikofunikira chifukwa kumatsimikiza kuphweka kwa integral yomwe yatuluka. Nthawi zambiri, timasankha \( u \) ngati ntchito yomwe imakhala yosavuta ikasinthidwa (\( du \)), ndi \( dv \) ngati ntchito yomwe imakhalabe yosavuta kuphatikiza.
Monga chitsogozo, tingagwiritse ntchito njira ya LIATE kusankha \( u \):
– Ntchito za Logarithmic (\( \ln (x) \))
– Ntchito zosinthira za trigonometric (\( \arctan(x), \arcsin(x), \arccos(x) \))
– Ntchito za lgebraic (\( x^n \))
– Ntchito za rigonometric za T (\( \sin(x), \cos(x) \))
– Ntchito zowonetsera (\( e^x \))
Ntchito yomwe imawonekera koyamba mu mndandanda wa LIATE nthawi zambiri imasankhidwa ngati \( u \).
2. Pezani \( u \) ndi Kuphatikiza \( dv \)
Mukasankha \( u \) ndi \( dv \), sitepe yotsatira ndikuwerengera zomwe zimachokera ku \( u \) (ndiko kuti, \( du \)) ndi integral ya \( dv \) (ndiko kuti, \( v \)).
3. Gwiritsani ntchito Fomula Yophatikizana Yochepa
Tikatha kuwerengera \( u \), \( du \), \( v \), ndi \( dv \), titha kugwiritsa ntchito njira yophatikizira pang'ono:
\[ \int u \, dv = uv – \int v \, du \]
4. Pezani ndi Kuphatikiza Zotsalira Zogwirizana
Gawo lomaliza ndi losavuta zotsatira zake ndikuphatikiza zotsalazo mpaka titapeza yankho lomaliza.
Zitsanzo Zothandiza
Chitsanzo 1: \( \int xe^x \, dx \)
Tiyerekeze kuti tikufuna kuphatikiza \( \int xe^x \, dx \).
1. Sankhani \( u \) ndi \( dv \):
– \( u = x \) (chifukwa zimakhala zosavuta zikachepetsedwa, kukhala 1)
– \( dv = e^x \, dx \) (popeza ndi yosavuta kuphatikiza, imakhalabe \( e^x \))
2. Pangani zinthu zochokera ku zinthu zina ndi zinthu zina:
– \( du = dx \)
– \( v = \int e^x \, dx = e^x \)
3. Gwiritsani ntchito njira yophatikizira pang'ono:
\[ \int xe^x \, dx = xe^x - \int e^x \, dx \]
4. Konzani gawo lotsalalo:
\[ \int e^x \, dx = e^x \]
Kotero:
\[ \int xe^x \, dx = xe^x - e^x + C \]
kapena
\[ \int xe^x \, dx = e^x (x – 1) + C \]
kumene \( C \) ndiye nthawi zonse yolumikizirana.
Chitsanzo 2: \( \int \ln(x) \, dx \)
Tiyerekeze kuti tikufuna kuphatikiza \( \int \ln(x) \, dx \).
1. Sankhani \( u \) ndi \( dv \):
– \( u = \ln(x) \) (chifukwa zimakhala zosavuta zikasiyanitsidwa)
– \( dv = dx \) (chifukwa ndi yosavuta kuphatikiza)
2. Pangani zinthu zochokera ku zinthu zina ndi zinthu zina:
– \( du = \frac{1}{x} \, dx \)
– \( v = \int dx = x \)
3. Gwiritsani ntchito njira yophatikizira pang'ono:
\[ \int \ln(x) \, dx = x \ln(x) – \int x \left(\frac{1}{x} \, dx\right) \]
\[ \int \ln(x) \, dx = x \ln(x) – \int 1 \, dx \]
4. Konzani gawo lotsalalo:
\[ \int 1 \, dx = x \]
Kotero:
\[ \int \ln(x) \, dx = x \ln(x) – x + C \]
Mavuto ndi Malangizo
Mavuto Ofala
1. Kusankha molakwika kwa \( u \) ndi \( dv \): Kusankha molakwika kwa \( u \) ndi \( dv \) kungapangitse kuti integral ikhale yovuta kwambiri. Kutsatira malangizo a LIATE nthawi zambiri kumathandiza.
2. Zosakaniza zotsala zovuta: Nthawi zina, mutatha kugwiritsa ntchito njira yophatikizira pang'ono, chosakaniza chotsalacho chimakhala chovutabe kuchithetsa. Pankhaniyi, kungakhale kofunikira kugwiritsanso ntchito njira yophatikiza pang'ono kapena kugwiritsa ntchito njira ina.
Nsonga
- Yesetsani ndi mitundu yosiyanasiyana ya ntchito kuti mumvetse mapatani ndikuwongolera luso lanu posankha \( u \) ndi \( dv \).
- Gwiritsani ntchito njira zosiyanasiyana zolumikizirana ngati pakufunika kutero, monga u-substitution.
– Musafulumire; yang'anani sitepe iliyonse kuti muwonetsetse kuti palibe zolakwika pakuchotsa ndi kuphatikiza.
Kutseka
Zophatikizana zosakwanira ndi chida champhamvu mu kuwerengera, zomwe zimatsegula njira yothetsera zophatikizana zovuta m'njira yosavuta. Mwa kumvetsetsa mfundo zoyambira ndi njira zoyenera zothetsera mavuto, komanso kuchita ndi zitsanzo zosiyanasiyana, titha kudziwa bwino njira iyi ndikuigwiritsa ntchito m'magawo osiyanasiyana a masamu ndi sayansi. Tikukhulupirira kuti bukuli lakuthandizani kumvetsetsa ndi kuthetsa molimba mtima zophatikizana zosakwanira.
