x = e t cos ( 3 t ) − 1 \displaystyle x = e ^{t} \cos ( \sqrt{3} t ) - 1 x = e t cos ( 3 t ) − 1 y = e t sin ( 3 t ) + 1 \displaystyle y = e ^{t} \sin ( \sqrt{3} t ) + 1 y = e t sin ( 3 t ) + 1 ( 0 ≤ t ≤ ln 7 ) ( 0 \leq t \leq \ln 7 ) ( 0 ≤ t ≤ ln 7 ) d x d t = e t cos 3 t − 3 e t sin 3 t \displaystyle \frac{dx}{dt} = e ^{t} \cos \sqrt{3} t - \sqrt{3} e ^{t} \sin \sqrt{3} t d t d x = e t cos 3 t − 3 e t sin 3 t d y d t = e t sin 3 t + 3 e t cos 3 t \displaystyle \frac{dy}{dt} = e ^{t} \sin \sqrt{3} t + \sqrt{3} e ^{t} \cos \sqrt{3} t d t d y = e t sin 3 t + 3 e t cos 3 t ∫ 0 ln 7 ( d x d t ) 2 + ( d y d t ) 2 \displaystyle \int _{0} ^{{\ln 7}} {\sqrt{\left( \frac{dx}{dt} \right) ^{2} + \left( \frac{dy}{dt} \right) ^{2}}} ∫ 0 l n 7 ( d t d x ) 2 + ( d t d y ) 2 = ∫ 0 ln 7 ( 1 + 3 ) e 2 t d x \displaystyle = \int _{0} ^{\ln 7} {\sqrt{( 1 + 3 ) e ^{2 t}} dx} = ∫ 0 l n 7 ( 1 + 3 ) e 2 t d x = [ 2 e t ] 0 ln 7 \displaystyle = \left[ \begin{array}{l} \begin{matrix} {} \\ {} \end{matrix} 2 e ^{t} \end{array} \right] _{0} ^{\ln 7} = [ 2 e t ] 0 l n 7 = 12 = 12 = 12