# What is the arclength of #f(t) = (t-e^(t),t-2/t+3)# on #t in [1,2]#?

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To find the arc length of the curve represented by the parametric equations (f(t) = (t - e^t, \frac{t - 2}{t + 3})) on the interval ([1, 2]), we use the formula for arc length:

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

where ( a = 1 ) and ( b = 2 ), and ( x(t) = t - e^t ) and ( y(t) = \frac{t - 2}{t + 3} ).

Now, let's find ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ):

[ \frac{dx}{dt} = 1 - e^t ] [ \frac{dy}{dt} = \frac{(t + 3)(1) - (t - 2)(1)}{(t + 3)^2} ]

Now, we plug these derivatives into the formula:

[ L = \int_{1}^{2} \sqrt{(1 - e^t)^2 + \left(\frac{(t + 3)(1) - (t - 2)(1)}{(t + 3)^2}\right)^2} , dt ]

[ L = \int_{1}^{2} \sqrt{(1 - e^t)^2 + \left(\frac{5}{(t + 3)^2}\right)^2} , dt ]

[ L = \int_{1}^{2} \sqrt{(1 - e^t)^2 + \frac{25}{(t + 3)^4}} , dt ]

This integral is not elementary and cannot be expressed in terms of elementary functions. Therefore, the arc length needs to be approximated using numerical methods such as Simpson's Rule or the Trapezoidal Rule.

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