Find the exact length of the parametric curve 0<t<pix = cos*e^ty= sin *e^t

Find the exact length of the parametric curve $0
$x=\mathrm{cos}\cdot {e}^{t}$
$y=\mathrm{sin}\cdot {e}^{t}$

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Laith Petty

Let y=f(x), $a\le x\le by=f\left(x\right),a\le x\le b$ be the given curve.The arc length L of such curve is given by the definite integral
$L={\int }_{a}^{b}\sqrt{1+{\left[{f}^{\prime }\left(x\right)\right)}^{2}}dx$
Let x=g(t), y=h(t) where $c\le x\le dc\le x\le d$ be the parametric equations of the curve y=f(x).
Then the arc length of the curve is given by
$L={\int }_{c}^{d}\sqrt{{\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}dt}$
Here $x\left(t\right)={\mathrm{cos}e}^{t},y\left(t\right)={\mathrm{sin}e}^{t}$ where $0 be the parametric equations of the curve y=f(x). Then the arc length of the curve is given by
$L={\int }_{0}^{\pi }sqt\left({\left({\mathrm{sin}e}^{t}\right)}^{2}+{\left({\mathrm{cos}e}^{t}\right)}^{2}dt\right)$
$={\int }_{0}^{\pi }sqt\left({\mathrm{sin}}^{2}{e}^{t}+{\mathrm{cos}}^{2}{e}^{t}dt\right)$
$={\int }_{0}^{\pi }dt$
$=\pi$