How would one proceed with proving that the integrals of motion
$$\text{mass} = \int_{-\infty}^\infty u(x,t)\mathrm{d}x$$
$$\text{torque} = \int_{-\infty}^\infty u^2(x,t)\mathrm{d}x$$
$$\text{energy} = \int_{-\infty}^{\infty}\bigg(\frac{1}{2}u^2_x(x,t) - u^3\bigg)\mathrm{d}x$$
are constants of motion (remain unchanged) for the Korteweg-De Vries (KdV) equation.
An idea would be straight differentiating with respect to time $t$ and then showing that it's equal to zero, thus meaning they are constant. But how would one carry out this differentiation properly ?