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.