Se have parametric curve with the formula for $x(t)$ and $y(t)$: $$x(t) = \int^{t}_{0} \sqrt{2cos(u) - 1}du$$ and $$ y(t)= \int^{t}_{0} \sqrt{2cos(u) + 5}du$$. I have to find the largest interval $[0, t_{0}]$, on which is this parametric curve defined and then I have to find also lenght of that curve. So how do I start this exercise, because first I tried to find the undefine integral of $x(t)$ and $y(t)$ but I could not find them.
Any help?
The integrand in $y(t)$ is always defined ($2\cos u+5\ge3$), but the expression under the square root in $x(t)$ becomes zero at $\cos u=\frac12$ or $u=\pi/3$, going upwards from $0$. Thus $t_0=\pi/3$ and the length of the curve is $$\int_0^{\pi/3}\sqrt{x'(t)^2+y'(t)^2}\,dt=\int_0^{\pi/3}\sqrt{(\sqrt{2\cos t-1})^2+(\sqrt{2\cos t+5})^2}\,dt$$ $$=\int_0^{\pi/3}\sqrt{4\cos t+4}\,dt=\int_0^{\pi/3}\sqrt{8\cos^2\frac t2}\,dt$$ $$=2\sqrt2\int_0^{\pi/3}\cos\frac t2\,dt=2\sqrt2\left[2\sin\frac t2\right]_0^{\pi/3}$$ $$=2\sqrt2\cdot2\sin\frac\pi6=2\sqrt2$$