We are tasked with evaluating the following limit:

$\lim _{x \to \infty} \left(\int_0^\pi \frac{t}{3 \cos^2(x \cdot t) + 1} \, dt\right)$

Step-by-step breakdown:

1. Expression Analysis: The integral involves a term $\cos^2(x \cdot t)$, which oscillates as $x \to \infty$. For large $x$, the cosine function rapidly oscillates for any non-zero $t$, and thus we expect the integral to become an average behavior over one period of oscillation.

2. Rough Intuition for the Integral:

   • The denominator $3 \cos^2(x \cdot t) + 1$ oscillates between values:
     • $3(0)^2 + 1 = 1$ when $\cos(x \cdot t) = 0$,
     • $3(1)^2 + 1 = 4$ when $\cos(x \cdot t) = \pm 1$.
   • As $x \to \infty$, the integral will smooth out because of the oscillations of $\cos(x \cdot t)$. The denominator's value will, on average, be dominated by the mean behavior of $\cos^2(x \cdot t)$.

3. Averaging the Function: The average value of $\cos^2(x \cdot t)$ over any interval of length $\pi$ is $\frac{1}{2}$, because: $\langle \cos^2(x \cdot t) \rangle = \frac{1}{\pi} \int_0^\pi \cos^2(x \cdot t) \, dt = \frac{1}{2}.$ Hence, for large $x$, the denominator can be approximated by: $3 \langle \cos^2(x \cdot t) \rangle + 1 = 3 \cdot \frac{1}{2} + 1 = \frac{3}{2} + 1 = \frac{5}{2}.$

4. Final Approximation: The integral for large $x$ can be approximated by: $\int_0^\pi \frac{t}{\frac{5}{2}} \, dt = \frac{2}{5} \int_0^\pi t \, dt.$ The integral of $t$ from 0 to $\pi$ is: $\int_0^\pi t \, dt = \frac{\pi^2}{2}.$ Therefore, the integral becomes: $\frac{2}{5} \cdot \frac{\pi^2}{2} = \frac{\pi^2}{5}.$

Conclusion:

Thus, the value of the limit is: $\lim _{x \to \infty} \left( \int_0^\pi \frac{t}{3 \cos^2(x \cdot t) + 1} \, dt \right) = \frac{\pi^2}{5}.$

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Would you like further details on any specific part of the solution? Here are some related questions for further exploration:

1. How do rapid oscillations affect integrals like this one?
2. What is the average value of trigonometric functions like $\cos^2(x \cdot t)$ over different intervals?
3. Can we apply a similar approach to other integrals with oscillatory components?
4. How do limits of integrals involving oscillatory functions behave when the frequency tends to infinity?
5. What techniques are commonly used for handling integrals involving rapidly oscillating terms?

Tip: When dealing with oscillatory integrals, averaging techniques can be particularly helpful for finding the long-term behavior as the frequency tends to infinity.