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Aniq integral va Nyuton–Leybnis formulasi
maktab darsligi: matematika 11-sinf, integral va uning tatbiqlari
Maqsad: Aniq integralni boshlang‘ich funksiya yordamida hisoblash.
Qoida va g‘oya
Aniq integral boshlang‘ich funksiya ayirmasi bilan topiladi (Nyuton–Leybnis formulasi): integral_a^b f(x)dx = F(b) - F(a). Bunda C yozilmaydi, chunki ayirmada yo‘qoladi. Xossalar: integral_a^b (f + g) = integral_a^b f + integral_a^b g; integral_a^a f = 0; integral_a^b f = -integral_b^a f; integral_a^b f = integral_a^c f + integral_c^b f. Toq funksiya uchun simmetrik oraliqda integral_-a^a f(x)dx = 0. Hisoblashda avval F ni yozing, so‘ng yuqori chegarani, keyin pastki chegarani qo‘ying va ayiring. Ayirish ishorasiga ehtiyot bo‘ling.
Namunaviy masalalar
integral_1^3 (2x + 1)dx ni hisoblang.
Yechim: F = x^2 + x. F(3) - F(1) = 12 - 2 = 10.
Javob: 10
Yechim: F = x^2 + x. F(3) - F(1) = 12 - 2 = 10.
Javob: 10
integral_0^pi sin x dx ni hisoblang.
Yechim: F = -cos x. -cos(pi) - (-cos 0) = 1 + 1 = 2.
Javob: 2
Yechim: F = -cos x. -cos(pi) - (-cos 0) = 1 + 1 = 2.
Javob: 2
integral_0^2 3x^2 dx ni hisoblang.
Yechim: F = x^3. 8 - 0 = 8.
Javob: 8
Yechim: F = x^3. 8 - 0 = 8.
Javob: 8
Tipik xatolar
- F(b) - F(a) o‘rniga F(a) - F(b) yozish.
- Pastki chegara manfiy bo‘lsa qavs ishlatmay ishorani yo‘qotish.
- Toq funksiya integralini simmetrik oraliqda nolga teng deb olishni bilmay, ortiqcha hisoblash.
Mashq
1
integral_1^4 sqrt(x) dx ni hisoblang.
14/3
2
integral_-1^2 x^2 dx ni hisoblang.
3
3
integral_0^(pi/2) cos x dx ni hisoblang.
1
4
integral_1^2 dx/x^2 ni hisoblang.
1/2