I must calculate the lenght of this curve $$\phi(t)=\big(\sin(2t),\cos(3t)\big)$$ so I must to calculate the integral of the norm of the derivative $$\left \| D\phi(t) \right \|=\sqrt{4\cos^2(2t)+9\sin^2(3t)}$$
$$\int _0^{2\pi} \sqrt{4\cos^2(2t)+9\sin^2(3t)}\, dt $$
I tried with various method, to change the function until they have the same arguments, but I can't solve this! Do you have some hints?
I found a solution.. It was easy in the end
$$\int _0^{2\pi} \sqrt{4\cos^2(2t)+9\sin^2(3t)}\, dt = \int _0^{2\pi} \sqrt{\frac{13}{2}}\sqrt{1 +\frac{4}{13}cos(4t)-\frac{2}{13}cos(6t)} dt=$$ $$=\sqrt{\frac{13}{2}}\int _0^{2\pi} 1 +\frac{1}{2}(\frac{4}{13}cos(4t)-\frac{2}{13}cos(6t))-\frac{1}{8}(\frac{4}{13}cos(4t)-\frac{2}{13}cos(6t))^2+.. dt\approx 15$$
More terms you develop better the approximation is..