$$ \int \frac{1}{x \ln (x+1) \ln ^{2}(x+2)} d x \sim \int \frac{1}{x \ln ^{3}(x+2)} d x $$ Endi yaqinlashuvni tasdiqlash uchun ushbu

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Berilgan integrallar orasidagi yaqinlashuvni tasdiqlash uchun, har ikki integrandning katta $$ x $$ qiymatlaridagi xatti-harakatini tahlil qilamiz.

**Integrallar:**
$$
\int \frac{1}{x \ln (x+1) \ln^{2}(x+2)} \, dx \quad \text{va} \quad \int \frac{1}{x \ln^{3}(x+2)} \, dx
$$

**1. Logarifmlarning yaqinlashuvi:**

Katta $$ x $$ uchun:
- $$ \ln(x+1) \approx \ln x $$
- $$ \ln(x+2) \approx \ln x $$

Shuning uchun:
$$
\ln(x+1) \approx \ln x \quad \text{va} \quad \ln^{2}(x+2) \approx \ln^{2}x
$$

**2. Integrandlarni yaqinlashtirish:**

Dastlabki integrand:
$$
\frac{1}{x \ln(x+1) \ln^{2}(x+2)} \approx \frac{1}{x \ln x \cdot \ln^{2} x} = \frac{1}{x \ln^{3}x}
$$

Ikkinchi integrand esa:
$$
\frac{1}{x \ln^{3}(x+2)} \approx \frac{1}{x \ln^{3}x}
$$

**3. Qo'shma natija:**

Katta $$ x $$ uchun har ikkala integrand ham taxminan $$ \frac{1}{x \ln^{3}x} $$ ga teng bo'lgani uchun, integrallar orasida quyidagi yaqinlashuv mavjud:
$$
\int \frac{1}{x \ln (x+1) \ln^{2}(x+2)} \, dx \sim \int \frac{1}{x \ln^{3}x} \, dx \quad \text{(katta $$ x $$ uchun)}
$$

Bu yaqinlashuv asosan $$ x $$ ning cheksizga yaqinlashishi holatida amal qiladi.
Katta $$ x $$ qiymatlarida, berilgan integrallar orasidagi yaqinlashuv $$ \int \frac{1}{x \ln (x+1) \ln^{2}(x+2)} \, dx \sim \int \frac{1}{x \ln^{3}x} \, dx $$ bo'lib, bu yaqinlashuv $$ x $$ ning cheksizga yaqinlashishi holatida amalga oshadi.

Endi yaqinlashuvni tasdiqlash uchun ushbu

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