I am reading "Calculus 4th Edition" by Michael Spivak.
There are the following problems and theorem in this book.
On p. 274 Problem 14.
If $f$ is integrable, then $$\int_a^b f(x) dx = \int_{a+c}^{b+c} f(x-c) dx.$$
On p. 275 Problem 16.
If $f$ is integrable, then $$\int_{ca}^{cb} f(t) dt = c \int_a^b f(ct) dt.$$
On p. 369 Theorem 2.
If $f$ and $g'$ are continuous, then $$\int_{g(a)}^{g(b)} f(u) du = \int_a^b f(g(x))\cdot g'(x) dx.$$
By Problems 14 and 16, if $f$ is integrable, then
$$\int_{ca+d}^{cb+d} f(u) du = \int_{ca}^{cb} f(u+d) du = c \int_a^b f(cx+d) dx.$$
Of course, if $f$ is continuous, the above equation holds by Theorem 2.
When $f$ is integrable, for what $g$ does $$\int_{g(a)}^{g(b)} f(u) du = \int_a^b f(g(x))\cdot g'(x) dx$$ hold?