https://file.aiquickdraw.com/mathbot/file/1725825433201d6iu9sqm.jpg

It seems like you've uploaded an image containing mathematical content. I will analyze it to provide an accurate interpretation and solution.

Give me a moment to extract the necessary information.The image contains the following mathematical expression:

\[
\frac{dx}{\sqrt{x^2 - 1}} = dt
\]

To solve this differential equation, let's proceed with these steps:

### 1. Integration of both sides:
Integrating both sides with respect to \(x\) and \(t\), respectively:

\[
\int \frac{dx}{\sqrt{x^2 - 1}} = \int dt
\]

The right side integrates simply to \(t + C\) (where \(C\) is the constant of integration). The left-hand side involves recognizing that the integral of \(\frac{1}{\sqrt{x^2 - 1}}\) is the inverse hyperbolic cosine function, \( \cosh^{-1}(x) \).

### 2. Result:
Thus, the solution is:

\[
\cosh^{-1}(x) = t + C
\]

Alternatively, you could express the solution as:

\[
x = \cosh(t + C)
\]

This is the general solution for the given equation.

Would you like further details or have any specific questions?

Here are 5 related questions to deepen your understanding:
1. How is the inverse hyperbolic cosine function defined?
2. What are the steps to solve integrals involving hyperbolic functions?
3. Can the same solution method be applied for different forms like \(\frac{1}{\sqrt{a

The query submitted by the user involves solving the differential equation \(\frac{dx}{\sqrt{x^2 - 1}} = dt\).

Learn how to solve the differential equation \(\frac{dx}{\sqrt{x^2 - 1}} = dt\) using integration techniques and inverse hyperbolic functions. This guide includes detailed steps and key concepts for advanced students in Grades 11-12 or university-level calculus.

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