Step 1: Understanding the Concept:
Observe the integrand. The numerator and denominator are identical: $e^{2025x} + e^{-2025x}$.
Step 2: Formula Derivation:
Any expression divided by itself is equal to 1 (provided it is non-zero).
$$\frac{e^{2025x} + e^{-2025x}}{e^{2025x} + e^{-2025x}} = 1$$
The integral becomes:
$$\int 1 \, dx$$
Step 3: Explanation:
The integral of 1 with respect to $x$ is simply $x + C$.
Note: The options provided seem to correspond to a different question (likely $\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} dx$). For the question as written, the answer is $x + C$.
Step 4: Final Answer:
The integral evaluates to $x + C$.