1. The problem is to evaluate the integral $$\int_0^1 \sqrt{y'^2 + t'^2} \, dx$$ where $y'$ and $t'$ represent derivatives with respect to $x$. 2. This integral represents the arc length of a curve parameterized by $y(x)$ and $t(x)$ from $x=0$ to $x=1$. 3. The formula for arc length $L$ of a curve given by functions $y(x)$ and $t(x)$ is: $$L = \int_a^b \sqrt{\left(\frac{dy}{dx}\right)^2 + \left(\frac{dt}{dx}\right)^2} \, dx$$ 4. To solve this integral, we need explicit expressions for $y(x)$ and $t(x)$ or their derivatives $y'$ and $t'$. Since these are not provided, the integral cannot be evaluated further. 5. If you provide the functions or their derivatives, I can help compute the integral step-by-step.