The integral of \(\tanh^2(x)\) with respect to \(x\) can be solved as follows:

We know that:
\[
\tanh^2(x) = 1 - \text{sech}^2(x)
\]

Thus, the integral becomes:
\[
\int \tanh^2(x) \, dx = \int \left(1 - \text{sech}^2(x)\right) \, dx
\]

Now, integrate each term separately:

1. The integral of \(1\) is simply \(x\).
\[
\int 1 \, dx = x
\]

2. The integral of \(-\text{sech}^2(x)\) is \(-\tanh(x)\), because the derivative of \(\tanh(x)\) is \(\text{sech}^2(x)\).
\[
\int -\text{sech}^2(x) \, dx = -\tanh(x)
\]

Thus, the complete integral is:
\[
\int \tanh^2(x) \, dx = x - \tanh(x) + C
\]
where \(C\) is the constant of integration.

Let me know if you want more details or have any questions!

Here are five related questions:
1. How do you derive the derivative of \(\tanh(x)\)?
2. What is the relationship between hyperbolic functions and their derivatives?
3. How does \(\tanh(x)\) compare to \(\sin(x)\) and \(\cos(x)\) in terms of periodicity?
4. How can you compute the integral of \(\tanh^3(x)\)?
5. What are the applications of hyperbolic functions in physics?

**Tip:** Hyperbolic functions are often used in problems involving hyperbolas, like in special relativity and electrical engineering, as analogs to circular trigonometric functions.

This guide provides a detailed solution for solving the integral of tanh^2(x), breaking down key steps such as using hyperbolic function identities and integration. Ideal for college-level calculus students studying hyperbolic functions and advanced integration techniques.

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