Evaluate the integral: integral coth² x dx
Evaluate the integral: $$ \int \coth^2 x \, dx. $$
Explanation of Method
The identity $\coth^2 x = 1 + \operatorname{csch}^2 x$ is the hyperbolic analogue of $\cot^2 x=\csc^2 x-1$, but note the sign: here the $1$ and the squared cofunction term are added rather than subtracted. This still reduces the integral to a constant term plus the standard derivative-of-coth form $\operatorname{csch}^2 x$.
- Rewrite $\coth^2 x$ 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 $\coth^2 x$
Recall the following identity for hyperbolic functions:
$$ \coth^2 x = 1 + \operatorname{csch}^2 x, $$
where $\operatorname{csch} x = \dfrac{1}{\sinh x}$ is the hyperbolic cosecant function. Using this identity, we can rewrite the integrand as:
$$ \coth^2 x = 1 + \operatorname{csch}^2 x. $$
Step 2 · Split the integral
Substitute the identity into the integral:
$$ \int \coth^2 x\,dx = \int (1 + \operatorname{csch}^2 x)\,dx. $$
Now, split the integral into two parts:
$$ \int \coth^2 x\,dx = \int 1\,dx + \int \operatorname{csch}^2 x\,dx. $$
Step 3 · Solve each integral
The first term is straightforward:
$$ \int 1\,dx = x. $$
The second term is a standard integral:
$$ \int \operatorname{csch}^2 x\,dx = -\coth x. $$
Step 4 · Combine the results
Combining the results from the two integrals, we get:
$$ \int \coth^2 x\,dx = x - \coth x + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Antiderivative |
|---|---|
| $1$ | $x$ |
| $\operatorname{csch}^2 x$ | $-\coth x$ |
Additional Notes
- The domain of $I$ excludes $x = 0$, since $\sinh x = 0$ there and $\coth x$, $\operatorname{csch}^2 x$ are undefined.
- Note the antiderivative of $\operatorname{csch}^2 x$ carries a minus sign, $-\coth x$, so even though the identity adds $\operatorname{csch}^2 x$, the final answer still subtracts $\coth x$.
- The result can be checked by differentiating $x - \coth x$ and confirming it returns $1 + \operatorname{csch}^2 x = \coth^2 x$.
Result
The solution to the integral is: