Question

What is the length of the arc of the curve y = ln (cos(x)) for 0 is less than or equal to x and x is less than or equal to pi over 3 ? (2 points)

A) the natural log of the quantity 1 plus the square root of 3
B) the natural log of the quantity 2 plus the square root of 3
C) the natural log of the quantity 3 plus the square root of 2
D) the natural log of the quantity 3 plus the square root of 3

Answer 1

d/dx ln(cosx) = -tanx

1 + (dy/dx)^2 = 1 + tan^2x = sec^2x
∫√sec^2x = ∫secx = secx tanx

all of those answers seem to indicate that you are doing
∫ tanx dx = -ln(cosx)
or something...

Did I miss something?

Answer 2

I think you interpreted it wrong. The answers are...

A) In ( 1+ square root of 3 )
B) In ( 2 + square root of 3 )
C) In ( 3 + square root of 2 )
D) In (3 + square root of 3 )

Answer 3

yes, I see that. But my integral has the value 2√3

so where did those logs come from?

Answer 4

[AI-generated answer — Explain Bot]

To find the length of the arc of a curve, we can use a formula called the arc length formula. The formula for finding the arc length of a function y = f(x) over the interval [a, b] is given by:

L = ∫[a, b] √(1 + (f'(x))^2) dx

In this case, we have the function y = ln(cos(x)) and the interval 0 ≤ x ≤ π/3.

To find the arc length, we need to calculate √(1 + (f'(x))^2) and then integrate it over the given interval.

First, let's find f'(x), the derivative of ln(cos(x)).
Using the chain rule:
f'(x) = 1/cos(x) * (-sin(x)) = -sin(x)/cos(x) = -tan(x)

Now, let's substitute f'(x) into the arc length formula:

L = ∫[0, π/3] √(1 + (-tan(x))^2) dx

Next, we simplify the integral:

L = ∫[0, π/3] √(1 + tan^2(x)) dx
= ∫[0, π/3] √(sec^2(x)) dx
= ∫[0, π/3] sec(x) dx

Integrating ∫sec(x) dx requires the use of a trigonometric identity. The identity we need here is:

∫sec(x) dx = ln|sec(x) + tan(x)| + C

where C is the constant of integration.

Applying this identity to our integral:

L = ln|sec(x) + tan(x)| |[0, π/3]

Now, we substitute the values π/3 and 0 into the integral:

L = ln|sec(π/3) + tan(π/3)| - ln|sec(0) + tan(0)|

Recall that sec(π/3) = 2 and tan(π/3) = √3. Also, sec(0) = 1 and tan(0) = 0.

L = ln|2 + √3| - ln|1 + 0|
= ln(2 + √3) - ln(1)
= ln(2 + √3)

Therefore, the length of the arc of the curve y = ln(cos(x)) for 0 ≤ x ≤ π/3 is ln(2 + √3).

So the correct answer is: A) the natural log of the quantity 1 plus the square root of 3.