Calculating Indefinite integral

Question

Suppose $f(x)=\int g(x)dx+C_1$,

Then how can I get $f(a-x)$?

I thought $f(a-x)=\int g(a-x)d(a-x)+C_1=-\int g(a-x)dx+C_1$. But it seems not correct.

Thanks.

Answer 1

Let $x=a-y\implies dx=-dy$

$$f(x)=\int g(x)dx+C_1$$

$$\implies f(a-y)=\int g(a-y)(-dy)+C_1=-\int g(a-y)dy+C_1$$