# Why replacing $\cos(x)$ by $e^{ix}$ in an integral and taking the real part doesn't always work? When does it work?

I wanted to solve the following integral using the residue theorem \begin{align*}\int_{0}^{2\pi} \frac{\cos(3x)}{5-4\cos(3x)} \,dx\end{align*} and for that I thought I could replace it by \begin{align*} \operatorname{Re}\biggl(\int_{0}^{2\pi} \frac{e^{3ix}}{5-4 e^{3ix}}\, dx\biggr) = \operatorname{Re}\biggl(\oint_\gamma \frac{z^{3}}{5-4 z^{3}} \frac{dz}{iz}\biggr)\end{align*} using $$z=e^{ix}$$ and $$\gamma$$ beeing a closed loop along the unit circle. Following this method, I could find that there are no singularities enclosed by the path so this should give me \begin{align*}\int_{0}^{2\pi} \frac{\cos(3x)}{5-4\cos(3x)}\, dx=0\end{align*} which is wrong. I thought the mistake is in the step \begin{align*}\int_{0}^{2\pi} \frac{\cos(3x)}{5-4\cos(3x)}\, dx=\operatorname{Re}\biggl(\int_{0}^{2\pi} \frac{e^{3ix}}{5-4 e^{3ix}}\, dx\biggr)\end{align*}. Indeed I calculated these on Wolfra alpha and didn't get the same result. I don't understand when I can use the trick of substituting an integral with $$\cos(x)$$ by the real part of an integral with $$e^{ix}$$ or substituting an integral with $$\sin(x)$$ by the imaginary part of an integral with $$e^{ix}$$ multiplied by $$i$$, since $$e^{ix}=\cos(x) + i\sin(x)$$. So, I would like to know in general when I am allowed to use it, and when I should be more careful about it.

Yep the mistake is there the problem is that the $$\operatorname{Re}$$ function is linear, but it doesn’t distribute over division. $$\operatorname{Re} \left(\frac{a}{b}\right)$$ is not the same as $$\frac{\operatorname{Re} (a)}{\operatorname{Re}(b)}.$$
Therefore, you cannot change $$\frac{\cos(3x)}{5-4\cos(3x)}$$ to $$\operatorname{Re}\left(\frac{e^{3ix}}{5-4 e^{3ix}}\right).$$
• for example, you can transform $e^x cos x$ to $e ^x e^{i x}$, which might be useful Feb 7, 2021 at 16:19
• but you cannot transform $cos x sin x$ to $e^{ix} * e^{ ix - \pi / 2}$ for example because you cannot change more than one factor Feb 7, 2021 at 16:21