18-Mavzu: Aniq integralda o‘zgaruvchini almashtirish va aniq integralni bo‘laklab integrallash.
Reja:
1. Bevosita integrallash
2.O’zgaruvchini almashtirish
3.Bo’laklab integrallash
Aniqmas integrallarni hisoblashda yangi o‘zgaruvchi kiritish usuli bilan soddaroq integralga erishib, ushbu
f(x)dx=f( (t))’(t)dt
munosabatdan foydalangan edik. Shunga o‘xshash masalani aniq integral uchun ham ko‘rib o‘taylik.
Aytaylik, f(x) funksiya [a;b] kesmada aniqlangan va uzluksiz bo‘lsin.
Teorema. Agar f(x) funksiya [a;b] da uzluksiz, x=(t) funksiya [;] kemada uzluksiz differensiallanuvchi, x=(t) funksiya qiymatlari to‘plami [a;b] kesmadan iborat hamda ()=a, ()=b bo‘lsa, u holda
= (3)
tenglik o‘rinli bo‘ladi.
Isboti. f(x) funksiya [a;b] da uzluksiz bo‘lgani uchun shu kesmada u boshlang‘ich funksiya F(x) ga ega. Shartga ko‘ra ()=a, ()=b bo‘lganligi sababli Nyuton-Leybnits formulasiga ko‘ra ![](data:image/png;base64,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)
Shuni ta’kidlash kerakki, aniq integralni o‘zgaruvchilarni almashtirish usuli bilan hisoblaganda integral ostidagi ifoda bilan bir qatorda integrallash chegaralari ham o‘zgaradi.
1-misol. hisoblang.
Yechish. Bu integralda x=sint almashtirishni bajaramiz. U holda x=sint funksiya yuqoridagi teoremadagi barcha shartlarni kesmada qanoatlantiradi va dx=costdt, a=0 da =0, b=1 da =/2. Demak, (3) formulaga ko‘ra
= .
2-misol. ni hisoblang.
Yechish. x=t2 deb o‘zgaruvchini almashtiramiz, u holda dx=2tdt va a=0 da t1==0, b=9 da t2==3 bo‘ladi. (3) formulaga ko‘ra
= .
3-misol. ni hisoblang.
Yechish. sinx=t deb almashtirish bajaramiz. U holda cosxdx=dt, t1=sin(/6)=1/2, t2=sin(/3)=/2 bo‘ladi. (3) formulaga asosan
= .
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