Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 13 / b / Solution

The factor $x^2$ is an even function and $\sin x$ is an odd function, so their product is an odd function. Its definite integral over the symmetric interval vanishes:Indeed the substitution rule $x=-t$ changes the integral over $[-\pi /2,0]$ into the negative of the integral over $[0,\pi /2]$.