https://file.aiquickdraw.com/mathbot/file/1726548492942nhsvzv1w.jpg

Let's break down the steps in this problem based on the provided image.

### Step 1: 
The integral to evaluate is:

\[
I = \int \text{sech}(u) \, du = \int \frac{1}{\cosh(u)} \, du
\]

We are instructed to use the hyperbolic substitution:

\[
t = \sinh(u)
\]

### Step 2:
To perform this substitution, we need to find \( \frac{dt}{du} \).

Given that \( t = \sinh(u) \), the derivative of \( \sinh(u) \) is:

\[
\frac{dt}{du} = \cosh(u)
\]

### Step 3:
The integral \( I \) can now be rewritten in terms of \( t \). We substitute \( \cosh(u) \, du = dt \) into the original integral:

\[
I = \int \frac{1}{\cosh(u)} \, du = \int \frac{1}{\cosh(u)} \cdot \cosh(u) \, dt = \int \frac{1}{t} \, dt
\]

### Step 4:
The integral becomes:

\[
I = \int \frac{1}{t} \, dt
\]

This is a standard integral, which evaluates to:

\[
I = \ln|t| + C
\]

### Step 5:
Since \( t = \sinh(u) \), we substitute back to express the result in terms of \( u \):

\[
I = \ln|\sinh(u)| + C
\]

Thus, the final result is:

\[
I = \ln|\sinh(u)| + C
\]

---

Would you like more details or have any questions about this solution?

Here are 5 related questions to deepen your understanding:

1. What are the properties of the hyperbolic functions \(\sinh(u)\) and \(\cosh(u)\)?
2. How would you evaluate \(\int \text{csch}(u) \, du\)?
3. What is the difference between trigonometric and hyperbolic substitutions in integrals?
4. Can the integral \(\int \cosh(u) \, du\) be solved using a similar substitution technique?
5. How does the substitution \(t = \sinh(u)\) simplify integrals involving \(\cosh(u)\)?

**Tip:** Understanding hyperbolic identities, like \(\cosh^2(u) - \sinh^2(u) = 1\), can make integrals involving hyperbolic functions easier to solve!

An alternative way to evaluate \(I = \int \text{sech}(u) \, du = \int \frac{1}{\cosh(u)} \, du\) is to use another hyperbolic trig substitution where \(t = \sinh(u)\) and \(\frac{dt}{du} = \cosh(u)\).

Explore the step-by-step solution to evaluate the integral of hyperbolic secant using a hyperbolic substitution, highlighting the transformation of the integral and the application of standard integral formulas. Suitable for university-level calculus.

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