Determine the indefinite integral. (Hint: Use an Identity.)
$\int \tanh^{2} x d x$
$\int \tanh^{2} x d x =$

Answer

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Answer

$\int \tanh^{2} x d x = - x - \tanh x + C$

Steps

Step 1 ：Given the integral $\int \tanh^{2} x d x$

Step 2 ：We use the identity $\tanh^{2} x = 1 - {sech}^{2} x$

Step 3 ：Substitute $\tanh^{2} x$ with $1 - {sech}^{2} x$ in the integral

Step 4 ：The integral becomes $\int (1 - {sech}^{2} x) d x$

Step 5 ：Separate the integral into two parts: $- \int d x - \int {sech}^{2} x d x$

Step 6 ：The integral of 1 with respect to x is x, and the integral of ${sech}^{2} x$ 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 ： $\int \tanh^{2} x d x = - x - \tanh x + C$