The image shows a problem asking to evaluate the integral:

$\int \frac{1}{\sqrt{\cosh{x} - 1}} \, dx$

The possible answers involve hyperbolic functions, logarithms, and constant terms. Here are the choices:

• a. $\frac{1}{\sqrt{2}} \ln \left| \csch \frac{x}{2} + \coth \frac{x}{2} \right|$
• b. $\frac{1}{\sqrt{2}} \ln \left| \csch \frac{x}{2} - \coth \frac{x}{2} \right|$
• c. $\frac{1}{\sqrt{2}} \ln \left| \csch \frac{x}{2} + \coth \frac{x}{2} \right|$
• d. $\frac{1}{\sqrt{2}} \ln \left| \csch \frac{x}{2} - \coth \frac{x}{2} \right|$

Solution outline:

1. Rewrite $\cosh{x} - 1$: Using the identity $\cosh{x} - 1 = 2\sinh^2{\left(\frac{x}{2}\right)}$, the integral becomes:

   $\int \frac{1}{\sqrt{2\sinh^2{\left(\frac{x}{2}\right)}}} \, dx = \frac{1}{\sqrt{2}} \int \csch{\left(\frac{x}{2}\right)} \, dx$

2. Solve the integral: The integral of $\csch{\left(\frac{x}{2}\right)}$ is known to be:

   $\ln \left| \csch \frac{x}{2} + \coth \frac{x}{2} \right|$

3. Final answer: Therefore, the correct answer is:

   $\boxed{a. \frac{1}{\sqrt{2}} \ln \left| \csch \frac{x}{2} + \coth \frac{x}{2} \right|}$

Would you like further explanation on any step?

Here are 5 related questions for further exploration:

1. What is the relationship between hyperbolic and trigonometric functions?
2. How does the substitution method simplify integrals involving hyperbolic functions?
3. What are some common integrals of hyperbolic functions?
4. Can $\csch(x)$ and $\coth(x)$ be expressed in terms of exponentials?
5. What are the key differences between $\cosh(x)$ and $\sinh(x)$ in integration problems?

Tip: When dealing with hyperbolic integrals, always check for identities that simplify the expression.