Answer
$\dfrac{19}{3}$
Work Step by Step
The length of the curve is given as: $L=\int_{0}^{\sqrt 5}\sqrt{r^2+(\dfrac{dr}{d\theta})^2}d\theta$
Thus, $L=\int_{0}^{\sqrt 5} \sqrt{(\theta^4+(2\theta))^2} d \theta$
Then, we have $L=\dfrac{1}{2}\int_{0}^{\sqrt 5} \sqrt {\theta^2+4}(2\theta d\theta)$
Plug $\theta^2+4 =p \implies dp=2\theta d\theta$
Then,we get $L =\dfrac{1}{2}\int_{4}^{9}p^{1/2} dp=\dfrac{1}{2}[\dfrac{2}{3}p^{3/2}]_{4}^{9} $
Thus, $L=\dfrac{27-8}{3}=\dfrac{19}{3}$
