The value of the integral ${\int }_{-\pi /2}^{\pi /2}(x\cos x)dx$ is:

Let $f(x)=x\cos x$.To check if the function is even or odd, replace $x$ with $-x$:
$$f(-x)=(-x)\cos (-x)$$
Since $\cos (-x)=\cos x$:
$$f(-x)=-x\cos x=-f(x)$$
Since $f(-x)=-f(x)$, the function is an odd function.According to the property of definite integrals, for an odd function:
$${\int }_{-a}^{a}f(x)dx=0$$
Therefore, ${\int }_{-\pi /2}^{\pi /2}(x\cos x)dx=0$.