Find the exact length of the curve.
y = ln(sec(x)),   0 ≤ x ≤ 𝜋/6

To find the exact length of the curve \( y = \ln(\sec(x)) \) over the interval \( 0 \leq x \leq \frac{\pi}{6} \), we use the formula for the arc length of a function \( y = f(x) \), which is given by:

\[
L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]

### Step 1: Find \(\frac{dy}{dx}\)
Given \( y = \ln(\sec(x)) \), we need to differentiate this with respect to \(x\):

\[
\frac{dy}{dx} = \frac{d}{dx} \left( \ln(\sec(x)) \right)
\]

Using the chain rule, we know that the derivative of \(\ln(u)\) is \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = \sec(x) \), and the derivative of \(\sec(x)\) is \(\sec(x)\tan(x)\). So:

\[
\frac{dy}{dx} = \frac{1}{\sec(x)} \cdot \sec(x)\tan(x) = \tan(x)
\]

### Step 2: Set up the arc length formula
Now, substitute \(\frac{dy}{dx} = \tan(x)\) into the arc length formula:

\[
L = \int_0^{\frac{\pi}{6}} \sqrt{1 + \tan^2(x)} \, dx
\]

Recall the trigonometric identity \( 1 + \tan^2(x) = \sec^2(x) \), so:

\[
L = \int_0^{\frac{\pi}{6}} \sqrt{\sec^2(x)} \, dx
\]

\[
L = \int_0^{\frac{\pi}{6}} \sec(x) \, dx
\]

### Step 3: Evaluate the integral
The integral of \(\sec(x)\) is well-known:

\[
\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C
\]

Thus, the arc length is:

\[
L = \left[ \ln|\sec(x) + \tan(x)| \right]_0^{\frac{\pi}{6}}
\]

### Step 4: Compute the values at the limits
At \( x = \frac{\pi}{6} \):

\[
\sec\left(\frac{\pi}{6}\right) = \frac{2}{\sqrt{3}}, \quad \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}
\]

So:

\[
\ln\left( \sec\left(\frac{\pi}{6}\right) + \tan\left(\frac{\pi}{6}\right) \right) = \ln\left( \frac{2}{\sqrt{3}} + \frac{1}{\sqrt{3}} \right) = \ln\left( \frac{3}{\sqrt{3}} \right) = \ln(\sqrt{3})
\]

At \( x = 0 \):

\[
\sec(0) = 1, \quad \tan(0) = 0
\]

So:

\[
\ln\left( \sec(0) + \tan(0) \right) = \ln(1) = 0
\]

### Step 5: Final result
The exact length of the curve is:

\[
L = \ln(\sqrt{3}) - 0 = \ln(\sqrt{3}) = \frac{1}{2} \ln(3)
\]

Thus, the exact length of the curve is:

\[
L = \frac{1}{2} \ln(3)
\]

Would you like further clarification or details on any step? Here are some related questions:

1. How do you differentiate logarithmic functions?
2. What is the trigonometric identity \(1 + \tan^2(x)\)?
3. How do you integrate \( \sec(x) \)?
4. What are the properties of the natural logarithm function?
5. How does the chain rule apply to logarithmic differentiation?

**Tip:** Knowing common integrals, such as the integral of \( \sec(x) \), can greatly speed up solving problems involving arc lengths.

Find the exact length of the curve. y = ln(sec(x)), 0 ≤ x ≤ 𝜋/6

Learn how to find the exact length of the curve y = ln(sec(x)) over the interval 0 ≤ x ≤ π/6. This solution explains step-by-step differentiation, the application of trigonometric identities, and the use of the arc length formula in calculus. Ideal for students in advanced high school or undergraduate calculus courses.

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