Solved on Feb 21, 2024

Find the length of the curve $y = \log\left(\frac{e^{x} - 1}{e^{x} + 1}\right)$ from $x = 1$ to $x = 2$ .

STEP 1

Assumptions
1. The curve is defined by the function $y = \log \left(\frac{e^{x} - 1}{e^{x} + 1}\right)$ .
2. We need to find the length of the curve from $x = 1$ to $x = 2$ .
3. The length of a curve defined by $y = f(x)$ from $x = a$ to $x = b$ is given by the integral $\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$ .

STEP 2

First, we need to find the derivative of $y$ with respect to $x$ , which is denoted as $\frac{dy}{dx}$ .

STEP 3

To find $\frac{dy}{dx}$ , we will apply the chain rule and the quotient rule of differentiation.

STEP 4

Let's denote the function inside the logarithm as $u(x) = \frac{e^{x} - 1}{e^{x} + 1}$ , so $y = \log(u(x))$ .

STEP 5

Using the chain rule, the derivative of $y$ with respect to $x$ is $\frac{dy}{dx} = \frac{1}{u(x)} \cdot \frac{du}{dx}$ .

STEP 6

Now we need to find $\frac{du}{dx}$ , which requires the quotient rule: $\frac{du}{dx} = \frac{v(x) \cdot \frac{du}{dx} - u(x) \cdot \frac{dv}{dx}}{[v(x)]^2}$ , where $u(x) = e^{x} - 1$ and $v(x) = e^{x} + 1$ .

STEP 7

Compute the derivatives $\frac{du}{dx}$ and $\frac{dv}{dx}$ . Since $u(x) = e^{x} - 1$ and $v(x) = e^{x} + 1$ , both derivatives are $e^{x}$ .

STEP 8

Apply the quotient rule:
$\frac{du}{dx} = \frac{(e^{x} + 1)(e^{x}) - (e^{x} - 1)(e^{x})}{(e^{x} + 1)^2}$

STEP 9

Simplify the numerator:
$\frac{du}{dx} = \frac{e^{2x} + e^{x} - e^{2x} + e^{x}}{(e^{x} + 1)^2}$

STEP 10

Combine like terms in the numerator:
$\frac{du}{dx} = \frac{2e^{x}}{(e^{x} + 1)^2}$

STEP 11

Now substitute $\frac{du}{dx}$ back into the derivative of $y$ :
$\frac{dy}{dx} = \frac{1}{u(x)} \cdot \frac{2e^{x}}{(e^{x} + 1)^2}$

STEP 12

Since $u(x) = \frac{e^{x} - 1}{e^{x} + 1}$ , we can substitute it back in:
$\frac{dy}{dx} = \frac{(e^{x} + 1)^2}{e^{x} - 1} \cdot \frac{2e^{x}}{(e^{x} + 1)^2}$

STEP 13

Simplify the expression by canceling out $(e^{x} + 1)^2$ :
$\frac{dy}{dx} = \frac{2e^{x}}{e^{x} - 1}$

STEP 14

Now we need to calculate $\left(\frac{dy}{dx}\right)^2$ :
$\left(\frac{dy}{dx}\right)^2 = \left(\frac{2e^{x}}{e^{x} - 1}\right)^2$

STEP 15

Next, we will set up the integral to find the length of the curve:
$\text{Length} = \int_{1}^{2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$

STEP 16

Substitute $\left(\frac{dy}{dx}\right)^2$ into the integral:
$\text{Length} = \int_{1}^{2} \sqrt{1 + \left(\frac{2e^{x}}{e^{x} - 1}\right)^2} \, dx$

STEP 17

Now we need to evaluate the integral. This may require numerical methods or a substitution if the integral is not elementary.

STEP 18

Let's first simplify the integrand:
$\sqrt{1 + \left(\frac{2e^{x}}{e^{x} - 1}\right)^2} = \sqrt{1 + \frac{4e^{2x}}{(e^{x} - 1)^2}}$

STEP 19

Combine the terms under the square root:
$\sqrt{1 + \frac{4e^{2x}}{(e^{x} - 1)^2}} = \sqrt{\frac{(e^{x} - 1)^2 + 4e^{2x}}{(e^{x} - 1)^2}}$

STEP 20

Simplify the numerator:
$\sqrt{\frac{(e^{x} - 1)^2 + 4e^{2x}}{(e^{x} - 1)^2}} = \sqrt{\frac{e^{2x} - 2e^{x} + 1 + 4e^{2x}}{(e^{x} - 1)^2}}$

STEP 21

Combine like terms:
$\sqrt{\frac{5e^{2x} - 2e^{x} + 1}{(e^{x} - 1)^2}}$

STEP 22

Now the integral becomes:
$\text{Length} = \int_{1}^{2} \sqrt{\frac{5e^{2x} - 2e^{x} + 1}{(e^{x} - 1)^2}} \, dx$

STEP 23

Because the integral does not have an elementary antiderivative, we must evaluate it numerically. This can be done using numerical integration techniques such as Simpson's rule, trapezoidal rule, or with the aid of a calculator or computer algebra system.

STEP 24

Use a numerical method to evaluate the integral:
$\text{Length} \approx \text{Numerical evaluation of the integral from } x = 1 \text{ to } x = 2$

STEP 25

After performing the numerical integration, we obtain the length of the curve.
The length of the curve $y = \log \left(\frac{e^{x} - 1}{e^{x} + 1}\right)$ from $x = 1$ to $x = 2$ is approximately equal to the result of the numerical integration.

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Find the length of the curve y=log⁡(ex−1ex+1)y = \log\left(\frac{e^{x} - 1}{e^{x} + 1}\right)y=log(ex+1ex−1​) from x=1x = 1x=1 to x=2x = 2x=2.

STEP 1

STEP 2

First, we need to find the derivative of y y y with respect to x x x, which is denoted as dydx \frac{dy}{dx} dxdy​.

STEP 3

To find dydx \frac{dy}{dx} dxdy​, we will apply the chain rule and the quotient rule of differentiation.

STEP 4

Let's denote the function inside the logarithm as u(x)=ex−1ex+1 u(x) = \frac{e^{x} - 1}{e^{x} + 1} u(x)=ex+1ex−1​, so y=log⁡(u(x)) y = \log(u(x)) y=log(u(x)).

STEP 5

Using the chain rule, the derivative of y y y with respect to x x x is dydx=1u(x)⋅dudx \frac{dy}{dx} = \frac{1}{u(x)} \cdot \frac{du}{dx} dxdy​=u(x)1​⋅dxdu​.

STEP 6

STEP 7

Compute the derivatives dudx \frac{du}{dx} dxdu​ and dvdx \frac{dv}{dx} dxdv​. Since u(x)=ex−1 u(x) = e^{x} - 1 u(x)=ex−1 and v(x)=ex+1 v(x) = e^{x} + 1 v(x)=ex+1, both derivatives are ex e^{x} ex.

STEP 8

Apply the quotient rule:dudx=(ex+1)(ex)−(ex−1)(ex)(ex+1)2 \frac{du}{dx} = \frac{(e^{x} + 1)(e^{x}) - (e^{x} - 1)(e^{x})}{(e^{x} + 1)^2} dxdu​=(ex+1)2(ex+1)(ex)−(ex−1)(ex)​

STEP 9

Simplify the numerator:dudx=e2x+ex−e2x+ex(ex+1)2 \frac{du}{dx} = \frac{e^{2x} + e^{x} - e^{2x} + e^{x}}{(e^{x} + 1)^2} dxdu​=(ex+1)2e2x+ex−e2x+ex​

STEP 10

Combine like terms in the numerator:dudx=2ex(ex+1)2 \frac{du}{dx} = \frac{2e^{x}}{(e^{x} + 1)^2} dxdu​=(ex+1)22ex​

STEP 11

Now substitute dudx \frac{du}{dx} dxdu​ back into the derivative of y y y:dydx=1u(x)⋅2ex(ex+1)2 \frac{dy}{dx} = \frac{1}{u(x)} \cdot \frac{2e^{x}}{(e^{x} + 1)^2} dxdy​=u(x)1​⋅(ex+1)22ex​

STEP 12

Since u(x)=ex−1ex+1 u(x) = \frac{e^{x} - 1}{e^{x} + 1} u(x)=ex+1ex−1​, we can substitute it back in:dydx=(ex+1)2ex−1⋅2ex(ex+1)2 \frac{dy}{dx} = \frac{(e^{x} + 1)^2}{e^{x} - 1} \cdot \frac{2e^{x}}{(e^{x} + 1)^2} dxdy​=ex−1(ex+1)2​⋅(ex+1)22ex​

STEP 13

Simplify the expression by canceling out (ex+1)2 (e^{x} + 1)^2 (ex+1)2:dydx=2exex−1 \frac{dy}{dx} = \frac{2e^{x}}{e^{x} - 1} dxdy​=ex−12ex​

STEP 14

Now we need to calculate (dydx)2 \left(\frac{dy}{dx}\right)^2 (dxdy​)2:(dydx)2=(2exex−1)2 \left(\frac{dy}{dx}\right)^2 = \left(\frac{2e^{x}}{e^{x} - 1}\right)^2 (dxdy​)2=(ex−12ex​)2

STEP 15

Next, we will set up the integral to find the length of the curve:Length=∫121+(dydx)2 dx \text{Length} = \int_{1}^{2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx Length=∫12​1+(dxdy​)2​dx

STEP 16

Substitute (dydx)2 \left(\frac{dy}{dx}\right)^2 (dxdy​)2 into the integral:Length=∫121+(2exex−1)2 dx \text{Length} = \int_{1}^{2} \sqrt{1 + \left(\frac{2e^{x}}{e^{x} - 1}\right)^2} \, dx Length=∫12​1+(ex−12ex​)2​dx

STEP 17

Now we need to evaluate the integral. This may require numerical methods or a substitution if the integral is not elementary.

STEP 18

Let's first simplify the integrand:1+(2exex−1)2=1+4e2x(ex−1)2 \sqrt{1 + \left(\frac{2e^{x}}{e^{x} - 1}\right)^2} = \sqrt{1 + \frac{4e^{2x}}{(e^{x} - 1)^2}} 1+(ex−12ex​)2​=1+(ex−1)24e2x​​

STEP 19

Combine the terms under the square root:1+4e2x(ex−1)2=(ex−1)2+4e2x(ex−1)2 \sqrt{1 + \frac{4e^{2x}}{(e^{x} - 1)^2}} = \sqrt{\frac{(e^{x} - 1)^2 + 4e^{2x}}{(e^{x} - 1)^2}} 1+(ex−1)24e2x​​=(ex−1)2(ex−1)2+4e2x​​

STEP 20

Simplify the numerator:(ex−1)2+4e2x(ex−1)2=e2x−2ex+1+4e2x(ex−1)2 \sqrt{\frac{(e^{x} - 1)^2 + 4e^{2x}}{(e^{x} - 1)^2}} = \sqrt{\frac{e^{2x} - 2e^{x} + 1 + 4e^{2x}}{(e^{x} - 1)^2}} (ex−1)2(ex−1)2+4e2x​​=(ex−1)2e2x−2ex+1+4e2x​​

STEP 21

Combine like terms:5e2x−2ex+1(ex−1)2 \sqrt{\frac{5e^{2x} - 2e^{x} + 1}{(e^{x} - 1)^2}} (ex−1)25e2x−2ex+1​​

STEP 22

Now the integral becomes:Length=∫125e2x−2ex+1(ex−1)2 dx \text{Length} = \int_{1}^{2} \sqrt{\frac{5e^{2x} - 2e^{x} + 1}{(e^{x} - 1)^2}} \, dx Length=∫12​(ex−1)25e2x−2ex+1​​dx

STEP 23

Because the integral does not have an elementary antiderivative, we must evaluate it numerically. This can be done using numerical integration techniques such as Simpson's rule, trapezoidal rule, or with the aid of a calculator or computer algebra system.

STEP 24

Use a numerical method to evaluate the integral:Length≈Numerical evaluation of the integral from x=1 to x=2 \text{Length} \approx \text{Numerical evaluation of the integral from } x = 1 \text{ to } x = 2 Length≈Numerical evaluation of the integral from x=1 to x=2

STEP 25

After performing the numerical integration, we obtain the length of the curve.The length of the curve y=log⁡(ex−1ex+1) y = \log \left(\frac{e^{x} - 1}{e^{x} + 1}\right) y=log(ex+1ex−1​) from x=1 x = 1 x=1 to x=2 x = 2 x=2 is approximately equal to the result of the numerical integration.

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Find the length of the curve $y = \log\left(\frac{e^{x} - 1}{e^{x} + 1}\right)$ from $x = 1$ to $x = 2$ .

First, we need to find the derivative of $y$ with respect to $x$ , which is denoted as $\frac{dy}{dx}$ .

To find $\frac{dy}{dx}$ , we will apply the chain rule and the quotient rule of differentiation.

Let's denote the function inside the logarithm as $u(x) = \frac{e^{x} - 1}{e^{x} + 1}$ , so $y = \log(u(x))$ .

Using the chain rule, the derivative of $y$ with respect to $x$ is $\frac{dy}{dx} = \frac{1}{u(x)} \cdot \frac{du}{dx}$ .

Compute the derivatives $\frac{du}{dx}$ and $\frac{dv}{dx}$ . Since $u(x) = e^{x} - 1$ and $v(x) = e^{x} + 1$ , both derivatives are $e^{x}$ .

Apply the quotient rule:
$\frac{du}{dx} = \frac{(e^{x} + 1)(e^{x}) - (e^{x} - 1)(e^{x})}{(e^{x} + 1)^2}$

Simplify the numerator:
$\frac{du}{dx} = \frac{e^{2x} + e^{x} - e^{2x} + e^{x}}{(e^{x} + 1)^2}$

Combine like terms in the numerator:
$\frac{du}{dx} = \frac{2e^{x}}{(e^{x} + 1)^2}$

Now substitute $\frac{du}{dx}$ back into the derivative of $y$ :
$\frac{dy}{dx} = \frac{1}{u(x)} \cdot \frac{2e^{x}}{(e^{x} + 1)^2}$

Since $u(x) = \frac{e^{x} - 1}{e^{x} + 1}$ , we can substitute it back in:
$\frac{dy}{dx} = \frac{(e^{x} + 1)^2}{e^{x} - 1} \cdot \frac{2e^{x}}{(e^{x} + 1)^2}$

Simplify the expression by canceling out $(e^{x} + 1)^2$ :
$\frac{dy}{dx} = \frac{2e^{x}}{e^{x} - 1}$

Now we need to calculate $\left(\frac{dy}{dx}\right)^2$ :
$\left(\frac{dy}{dx}\right)^2 = \left(\frac{2e^{x}}{e^{x} - 1}\right)^2$

Next, we will set up the integral to find the length of the curve:
$\text{Length} = \int_{1}^{2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$

Substitute $\left(\frac{dy}{dx}\right)^2$ into the integral:
$\text{Length} = \int_{1}^{2} \sqrt{1 + \left(\frac{2e^{x}}{e^{x} - 1}\right)^2} \, dx$

Let's first simplify the integrand:
$\sqrt{1 + \left(\frac{2e^{x}}{e^{x} - 1}\right)^2} = \sqrt{1 + \frac{4e^{2x}}{(e^{x} - 1)^2}}$

Combine the terms under the square root:
$\sqrt{1 + \frac{4e^{2x}}{(e^{x} - 1)^2}} = \sqrt{\frac{(e^{x} - 1)^2 + 4e^{2x}}{(e^{x} - 1)^2}}$

Simplify the numerator:
$\sqrt{\frac{(e^{x} - 1)^2 + 4e^{2x}}{(e^{x} - 1)^2}} = \sqrt{\frac{e^{2x} - 2e^{x} + 1 + 4e^{2x}}{(e^{x} - 1)^2}}$

Combine like terms:
$\sqrt{\frac{5e^{2x} - 2e^{x} + 1}{(e^{x} - 1)^2}}$

Now the integral becomes:
$\text{Length} = \int_{1}^{2} \sqrt{\frac{5e^{2x} - 2e^{x} + 1}{(e^{x} - 1)^2}} \, dx$

Use a numerical method to evaluate the integral:
$\text{Length} \approx \text{Numerical evaluation of the integral from } x = 1 \text{ to } x = 2$

After performing the numerical integration, we obtain the length of the curve.
The length of the curve $y = \log \left(\frac{e^{x} - 1}{e^{x} + 1}\right)$ from $x = 1$ to $x = 2$ is approximately equal to the result of the numerical integration.