Let's say I have a function that I want to integrate an even function multiplied by $i\sin(x)$ between $-1$ and $1$ and the function is $1-|x|$ then does this integral become zero because integrating an odd function over a symmetric domain is zero or is $i\sin(x)$ not odd anymore when multiplied by $i$.
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$\begingroup$Multiplying an odd or even function by a constant doesn't change the parity of the function.
The symbol $i$ sometimes represents the imaginary unit, which is a number, a constant.
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