why $2\int_{0}^{1}xe^{iux^2}dx= \operatorname{sinc}(\frac{u}{2})$

Question

$2\int_{0}^{1}xe^{iux^2}dx= \operatorname{sinc}(\frac{u}{2})= \frac{\sin(u/2)}{u/2}$ When reading some papers I always meet this kind of expansion. But I cannot understand it. May anyone here help me please?

Answer 1

$$I=2\int_0^1xe^{iux^2}dx$$ Now make the substitution $v=iux^2$ so $dx=\frac{dv}{2iux}$ giving: $$I=\frac{1}{iu}\int_0^{iu}e^vdv=\frac{e^{iu}-1}{iu}=\frac{\sin(u)-i\cos(u)+i}{u}$$