Substituting an even function for an odd function in an integral In a book I am reading it says while evaluating an integral it says you cannot substitute an odd function for an even function of another variable ( for example substituting sinx with cosu and integrating with respect to u) if the domain of integration of original integral contains 0.

Why is this so? Any help would be appreciated.
Hint:

For: $0&lt;a&lt;b$ we have:
$$
\int_{-a}^b f(x)dx=\int_{-a}^a f(x)dx+\int_{a}^b f(x)dx
$$
If $f(-x)=-f(x)$ is odd 
$$
\int_{-a}^a f(x)dx=\int_{-a}^0 f(x)dx +\int_{0}^a f(x)dx=\int_{0}^a f(-x)dx+\int_{0}^a f(x)dx=0
$$

but this is no true (in general) if $f(x)$ is not an odd function.
I think the paper mentions the reason as well:


  ... because
  in that case a substitution by an even function would not work since
  an even function cannot assume both positive and negative values.