Substituting $u = x^{2}$ into $\int 2x\cos(x^{2})\,dx$ gives $\int \cos u\,du$, not $\int \cos u\,dx$. Why must the differential change too?

B. Because du = 2x dx replaces both the factor and the dx

Substitution is a change of variable throughout, integrand and differential together. The $2x$ is consumed precisely by converting $dx$ into $du$.