Evaluate the integral: integral tanh² x dx
Evaluate the integral: $$ \int \tanh^2 x \, dx. $$
Explanation of Method
The hyperbolic identity $\tanh^2 x = 1 - \operatorname{sech}^2 x$ plays the same role for hyperbolic tangent that $\tan^2 x=\sec^2 x-1$ plays for ordinary tangent. It converts a power with no direct antiderivative into a difference of a constant term and the standard derivative-of-tanh form $\operatorname{sech}^2 x$.
- Rewrite $\tanh^2 x$ using the hyperbolic Pythagorean identity.
- Split the integral into two simpler integrals.
- Integrate each term using standard results.
- Combine the results and add the constant of integration.
Step 1 · Use the identity for $\tanh^2 x$
Recall the following identity for hyperbolic functions:
$$ \tanh^2 x = 1 - \operatorname{sech}^2 x, $$
where $\operatorname{sech} x = \dfrac{1}{\cosh x}$ is the hyperbolic secant function. Using this identity, we can rewrite the integrand as:
$$ \tanh^2 x = 1 - \operatorname{sech}^2 x. $$
Step 2 · Split the integral
Substitute the identity into the integral:
$$ \int \tanh^2 x\,dx = \int (1 - \operatorname{sech}^2 x)\,dx. $$
Now, split the integral into two parts:
$$ \int \tanh^2 x\,dx = \int 1\,dx - \int \operatorname{sech}^2 x\,dx. $$
Step 3 · Solve each integral
The first term is straightforward:
$$ \int 1\,dx = x. $$
The second term is a standard integral:
$$ \int \operatorname{sech}^2 x\,dx = \tanh x. $$
Step 4 · Combine the results
Combining the results from the two integrals, we get:
$$ \int \tanh^2 x\,dx = x - \tanh x + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Antiderivative |
|---|---|
| $1$ | $x$ |
| $-\operatorname{sech}^2 x$ | $-\tanh x$ |
Additional Notes
- The integrand is defined for all real $x$, since $\cosh x \neq 0$ everywhere.
- Compare with the ordinary trig case: $\tan^2 x = \sec^2 x - 1$ gives $\tan x - x + C$, while here the identity is flipped ($\tanh^2 x = 1-\operatorname{sech}^2 x$), producing $x-\tanh x+C$ — note the term order swaps.
- The result can be checked by differentiating $x - \tanh x$ and confirming it returns $1 - \operatorname{sech}^2 x = \tanh^2 x$.
Result
The solution to the integral is: