Should substitution in indefinite integral be monotone?

Question

If me evaluate a definite integral via substitution, there is often the requirement that the substitution function should be monotone (to adjust the limits), is this also the case when computing an indefinite integral? If yes, is there a counterexample?

Answer 1

Yes. When you substitute, you need to use a strictly increasing (or decreasing) function. For example consider $\int x \; dx$ and substitute $x=c$ (a constant). This is clearly not strictly monotone increasing and $$ \int x \; dx=\frac{x^2}{2}+C \neq cx+C'=\int c \; dx $$