To evaluate the expression

$\lim_{n \to \infty} n^2 \cdot \int_{n}^{n+1} \ln\left(1 + \frac{x}{1 + x^3}\right) \, dx$

let's proceed with analyzing the behavior of the integrand and applying some asymptotic analysis as $n \to \infty$.

Step 1: Analyzing the Integrand

We have: $\ln\left(1 + \frac{x}{1 + x^3}\right).$ As $x \to \infty$, the term $\frac{x}{1 + x^3}$ tends to zero since the denominator $1 + x^3$ grows much faster than the numerator $x$. Hence, we can use the approximation $\ln(1 + u) \approx u$ for $u$ close to zero.

Thus, for large $n$, we can approximate: $\ln\left(1 + \frac{x}{1 + x^3}\right) \approx \frac{x}{1 + x^3}.$

Step 2: Approximating the Integral

Now, we want to evaluate: $\int_{n}^{n+1} \frac{x}{1 + x^3} \, dx.$ For large $x$, we approximate $1 + x^3 \approx x^3$, so $\frac{x}{1 + x^3} \approx \frac{x}{x^3} = \frac{1}{x^2}.$ Thus, the integral becomes approximately: $\int_{n}^{n+1} \frac{1}{x^2} \, dx.$

Step 3: Evaluating the Approximate Integral

Now we compute: $\int_{n}^{n+1} \frac{1}{x^2} \, dx = \left[-\frac{1}{x}\right]_{n}^{n+1} = -\frac{1}{n+1} + \frac{1}{n}.$ Simplifying, we get: $\int_{n}^{n+1} \frac{1}{x^2} \, dx \approx \frac{1}{n} - \frac{1}{n+1}.$

Using the approximation $\frac{1}{n} - \frac{1}{n+1} \approx \frac{1}{n^2}$ for large $n$, the integral is approximately $\frac{1}{n^2}$.

Step 4: Taking the Limit

Now we have: $\lim_{n \to \infty} n^2 \cdot \int_{n}^{n+1} \ln\left(1 + \frac{x}{1 + x^3}\right) \, dx \approx \lim_{n \to \infty} n^2 \cdot \frac{1}{n^2} = \lim_{n \to \infty} 1 = 1.$

Final Answer

$\lim_{n \to \infty} n^2 \cdot \int_{n}^{n+1} \ln\left(1 + \frac{x}{1 + x^3}\right) \, dx = 1.$

Would you like more details on the steps or have any further questions?

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Here are some related questions that expand on this topic:

1. How can we generalize this approach for other integrals with similar limits?
2. What other approximation techniques could be applied for different asymptotic behavior?
3. How does the choice of approximation influence the result in limit calculations?
4. Could we approach this problem using numerical methods? If so, how?
5. What is the role of the Taylor series in such asymptotic limit problems?

Tip: When evaluating limits of integrals as $n \to \infty$, approximating the integrand based on its behavior for large $n$ can often simplify the calculation considerably.