<|begin_of_text|><|start_header_id|>system<|end_header_id|>

You are a helpful chatbot that always answers in Uzbek.<|eot_id|><|start_header_id|>user<|end_header_id|>

5x^2 ning integrali qanday?<|eot_id|><|start_header_id|>assistant<|end_header_id|>

Agar siz $x$$\int 5x^2 = \frac{5}{3}x^3 + C$$ ga nisbatan integralni nazarda tutgan bo'lsangiz, unda $C$ doimiy.<|eot_id|><|start_header_id|>user<|end_header_id|>

Xy^2 + 3y + z$ ning $y$ ga nisbatan integrali qanday?<|eot_id|><|start_header_id|>assistant<|end_header_id|>

Integratsiya uchun y ga nisbatan x va z ni doimiylar sifatida qabul qilamiz va integratsiyaning kuch qoidasidan foydalanamiz. Shunday qilib, bizda: ∫ [(2x) /(2+3y+z) dy + C bu yerda C integratsiya konstantidir. Birinchi hadni integratsiya qilish uchun biz almashtirishdan foydalanishimiz mumkin. U = 2+3y+z, keyin du/dy = 3, bu dy = (1/3) du degan ma'noni anglatadi. U va dy ni integralda almashtirish orqali biz: (2x/3) ∫ u^-1) du + C = (2x/3) ∫ ln + C = (2x/3) ln + 2y+z + C Shuning uchun (2x/(2+2y+z)) + (xy^2) + (y^2) + 3y+z ning y ga nisbatan integratsiyasi: (2x/3) ln + 2y+z + (<|eot_id|>