can u show me the solution for getting the inverse of cschx

The inverse of the hyperbolic cosecant function, \( \text{csch}(x) \), can be derived using the following process:

### Step-by-Step Solution for Finding the Inverse of \( \text{csch}(x) \)

We start with the equation:

\[
y = \text{csch}(x)
\]

The hyperbolic cosecant is defined as:

\[
y = \frac{1}{\sinh(x)}
\]

Now, solve for \( x \) in terms of \( y \):

\[
y = \frac{1}{\sinh(x)} \implies \sinh(x) = \frac{1}{y}
\]

Next, recall that the inverse of \( \sinh(x) \) is \( \text{arsinh}(x) \), which is expressed as:

\[
x = \text{arsinh}\left(\frac{1}{y}\right)
\]

The formula for the inverse hyperbolic sine (arsinh) is:

\[
\text{arsinh}(x) = \ln(x + \sqrt{x^2 + 1})
\]

So, we substitute \( \frac{1}{y} \) into the inverse hyperbolic sine formula:

\[
x = \ln\left(\frac{1}{y} + \sqrt{\left(\frac{1}{y}\right)^2 + 1}\right)
\]

Simplify the expression:

\[
x = \ln\left(\frac{1}{y} + \sqrt{\frac{1}{y^2} + 1}\right)
\]

\[
x = \ln\left(\frac{1}{y} + \frac{\sqrt{1 + y^2}}{|y|}\right)
\]

This gives the inverse of the hyperbolic cosecant function:

\[
\text{csch}^{-1}(y) = \ln\left(\frac{1}{y} + \frac{\sqrt{1 + y^2}}{|y|}\right)
\]

Thus, the inverse of \( \text{csch}(x) \) is:

\[
\boxed{\text{csch}^{-1}(x) = \ln\left(\frac{1}{x} + \frac{\sqrt{1 + x^2}}{|x|}\right)}
\]

Let me know if you'd like further details!

### Related Questions:

1. What is the inverse of the hyperbolic cosine \( \cosh^{-1}(x) \)?
2. How do you derive the inverse of \( \sinh(x) \)?
3. What are the properties of the hyperbolic functions?
4. Can you solve an equation involving \( \text{csch}(x) \) and its inverse?
5. What are the applications of hyperbolic functions in physics?

**Tip:** The logarithmic form of inverse hyperbolic functions makes them useful in solving integrals and differential equations involving hyperbolic terms.

Learn how to derive the inverse of the hyperbolic cosecant function csch(x) with a detailed step-by-step solution. This problem involves key mathematical concepts like hyperbolic functions, inverse functions, and logarithms. Ideal for students in advanced calculus courses.

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