Interchanging integral with real and imaginary operators?

$$ int sin(3x) cos(nx) dx to Re int sin(3x) e^{inx} dx$$

$$ sin(3x) to Im e^{i3x}$$

Hence,

$$ Re left( Im int e^{i(3+n)x} dx right) $$

Or,

$$ Re left(Im frac{e^{i(3+n)}}{i(3+n)} right) to Re left(Im frac{-ie^{i(3+n)}}{(3+n)} right)$$

Considering,

$$ Im frac{-ie^{i(3+n)}}{(3+n)} to -frac{cos(3+n)}{3+n}$$

Hence,

$$ int sin(3x) cos(nx) dx = – frac{ cos(3+n)}{3+n}$$

Now this is wrong.. why?
Btw I am using result from here