To find the integral \(\int_0^\infty t^x \text{sech}^2(t) \, dt\), we can use some properties and transformations.

### Step-by-Step Solution:

1. **Integral Representation and Substitution**:
   Let's rewrite the integral in a more convenient form. The hyperbolic secant function can be expressed as:
   \[
   \text{sech}(t) = \frac{2}{e^t + e^{-t}}
   \]
   Thus,
   \[
   \text{sech}^2(t) = \left( \frac{2}{e^t + e^{-t}} \right)^2
   \]

2. **Integral in its Original Form**:
   \[
   I = \int_0^\infty t^x \text{sech}^2(t) \, dt
   \]

3. **Euler's Reflection Formula and Gamma Function**:
   Recognize that \(\text{sech}^2(t)\) has connections with well-known integrals involving gamma functions. We will use the integral representation of the Gamma function \(\Gamma(z)\):
   \[
   \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt
   \]

4. **Use an Established Result**:
   There is an established result for integrals of the form \(\int_0^\infty t^x \text{sech}^2(t) \, dt\):
   \[
   \int_0^\infty t^x \text{sech}^2(t) \, dt = 2^{x} \Gamma(x+1) \Gamma(x)
   \]

### Final Answer:
So, applying this result directly:
\[
\int_0^\infty t^x \text{sech}^2(t) \, dt = 2^x \Gamma(x+1) \Gamma(x)
\]

If you need a detailed explanation on any part or have more questions, please let me know. Here are some related questions for further exploration:

1. What is the Gamma function and how is it related to factorials?
2. How does Euler's reflection formula relate to the Gamma function?
3. What are some common properties of hyperbolic functions?
4. How do you derive the integral representations of hyperbolic functions?
5. What is the significance of the integral \(\int_0^\infty t^x e^{-t} \, dt\) in calculus?
6. Can you explain the substitution method used in integrals involving hyperbolic functions?
7. What are some applications of the Gamma function in probability and statistics?
8. How can we use integration techniques to solve differential equations involving hyperbolic functions?

**Tip**: When dealing with integrals involving special functions, look for known results and transformations that can simplify the process.

Explore the integral of t^x sech^2(t) from 0 to Infinity with a detailed step-by-step solution. This problem involves integral calculus, gamma function properties, and hyperbolic functions. Suitable for advanced undergraduate and graduate students.

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