Step 1 :Given the integral \(\int \tanh^{2} x dx\)
Step 2 :We use the identity \(\tanh^{2} x = 1 - \text{sech}^{2} x\)
Step 3 :Substitute \(\tanh^{2} x\) with \(1 - \text{sech}^{2} x\) in the integral
Step 4 :The integral becomes \(\int (1 - \text{sech}^{2} x) dx\)
Step 5 :Separate the integral into two parts: \(-\int dx - \int \text{sech}^{2} x dx\)
Step 6 :The integral of 1 with respect to x is x, and the integral of \(\text{sech}^{2} x\) with respect to x is \(\tanh x\)
Step 7 :So, the integral becomes \(-x - \tanh x + C\), where C is the constant of integration
Step 8 :\(\boxed{\int \tanh^{2} x dx = -x - \tanh x + C}\)