114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).
a. Find the length of the curve from x = 1 to x = a and call it L(a).
(Hint: The change of variables u = sqrt(x^2 + 1) allows evaluation by partial fractions.)

Verified step by step guidance

1. Recall the formula for the arc length of a curve defined by y = f(x) from x = a to x = b: \[L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]
2. For the curve y = \(\ln\)(x), compute the derivative: \[\frac{dy}{dx} = \frac{1}{x}\]
3. Substitute the derivative into the arc length formula: \[L(a) = \int_1^a \sqrt{1 + \left(\frac{1}{x}\right)^2} \, dx = \int_1^a \sqrt{1 + \frac{1}{x^2}} \, dx = \int_1^a \sqrt{\frac{x^2 + 1}{x^2}} \, dx\]
4. Simplify the integrand: \[L(a) = \int_1^a \frac{\sqrt{x^2 + 1}}{x} \, dx\]
5. Use the substitution suggested: let \[u = \sqrt{x^2 + 1}\] Then express dx and x in terms of u to rewrite the integral and evaluate it using partial fractions.

Key Concepts

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is given by the integral L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula measures the distance along the curve by summing infinitesimal line segments, requiring the derivative of the function to account for the slope.

Derivative of the Natural Logarithm Function

For y = ln(x), the derivative dy/dx = 1/x. This derivative is essential in the arc length formula to find the integrand √(1 + (1/x)^2) = √(1 + 1/x^2), which simplifies the integral needed to compute the curve's length.

Integration Techniques: Substitution and Partial Fractions

To evaluate the arc length integral, substitution (u = √(x^2 + 1)) simplifies the integrand, transforming it into a rational function. Partial fraction decomposition then breaks this rational function into simpler terms, making the integral solvable using standard methods.