questions in understanding the textbook about fourier transform

Question

I am confused about if the text book set $z = x-y$, then why there is no $-$ after variable change? I think it is obvious that $dy = - dz$

Answer 1

You are right there is a minus sign but also the boundaries of the integral will change after the change of variables, so the integral will be

$-\int_{+\infty}^{-\infty}=\int_{-\infty}^{+\infty}= \int_{\mathbb R}$

Answer 2

They have compensated for this negative sign, but it's hidden in their $\int_R$ notation. Substitution affects both the $d$ term and the bounds of an integral, but we can't see that here. Writing it with explicit bounds, and inserting an intermediate calculation, we get $$ \int_{-\infty}^\infty [\cdots]dy=-\int_\infty^{-\infty}[\cdots]dz\\ =\int_{-\infty}^\infty [\cdots ]dz $$ Hopefully this is clearer.