I am asked the following:
Evaluate $\int_{0}^{2} f(x-2) \ dx$ knowing that $\int_{-2}^{0} f(x) \ dx = 3$
When I try to make a u-substitution I get
$$ u=x-2\\ du = dx\\ \\ \text{when } x = -2 \Rightarrow u = -4\\ \text{when } x = 0 \Rightarrow u = -2 $$
which leads me nowhere. Is there a typing error on the exercise? Would it make more sense if the original question was the following?
Evaluate $\int_{0}^{2} f(x+2) \ dx$ knowing that $\int_{-2}^{0} f(x) \ dx = 3$
in which we would have
$$ u=x+2\\ du = dx\\ \\ \text{when } x = -2 \Rightarrow u = 0\\ \text{when } x = 0 \Rightarrow u = 2 $$
so that
$$\int_{0}^{2} f(x+2) \ dx = \int_{-2}^{0} f(x) \ dx = 3$$
Thank you.
The subsitution $u=x-2$ doesn't lead you no where, it gives you your answer! With this substitution, your bounds are $-2\leq u \leq 0$ and you get $$ \int_0^2f(x-2)\,dx=\int_{-2}^0 f(u)\,du=\int_{-2}^0f(x)\,dx=3 $$ The variable of integration is just a dummy variable, it is irrelevant.