Integrating the inverse hyperbolic functions \(\text{arsinh}(x)\), \(\text{arcosh}(x)\), and \(\text{artanh}(x)\) can be done using standard integration techniques. Below are the integrals of each of these functions:

### 1. Integral of the Inverse Hyperbolic Sine (\(\text{arsinh}(x)\))
\[
\int \text{arsinh}(x) \, dx = x \cdot \text{arsinh}(x) - \sqrt{x^2 + 1} + C
\]
where \(C\) is the constant of integration.

### 2. Integral of the Inverse Hyperbolic Cosine (\(\text{arcosh}(x)\))
\[
\int \text{arcosh}(x) \, dx = x \cdot \text{arcosh}(x) - \sqrt{x^2 - 1} + C
\]
where \(C\) is the constant of integration.

### 3. Integral of the Inverse Hyperbolic Tangent (\(\text{artanh}(x)\))
\[
\int \text{artanh}(x) \, dx = x \cdot \text{artanh}(x) + \frac{1}{2} \ln(1-x^2) + C
\]
where \(C\) is the constant of integration.

These integrals are derived using integration by parts, and recognizing standard integrals.

Do you need more detailed steps on how these integrals are derived, or have any other questions?

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Here are five related questions to consider:

1. How can integration by parts be applied to derive these integrals?
2. What are the applications of integrating inverse hyperbolic functions in physics or engineering?
3. How do these integrals compare to the integrals of the inverse trigonometric functions?
4. What are the definite integrals of these functions over specific intervals?
5. How can the integrals of inverse hyperbolic functions be used to solve real-world problems?

**Tip:** When integrating inverse functions, consider using integration by parts, where you can set the inverse function as the first function to differentiate.

Explore the integrals of inverse hyperbolic functions: arsinh(x), arcosh(x), and artanh(x). Learn their formulas and how to derive them using integration by parts. Suitable for college-level mathematics.

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