Calculate Integral Using Fourier Series?

Question

I got this integral that I have been asked to calculate: $\int_{0}^{2\pi} |3+4e^{10ix}+5e^{100ix}|^{2}dx$

I tried using Parseval's identity and tried to convert it to Fourier series. I think there is an easy way to solve it that I am missing.

Thanks

Answer 1

There is a much easier way: just multiply the integrand out.

$$\begin{align}|3+4e^{10ix}+5e^{100ix}|^{2} &= (3+4e^{10ix}+5e^{100ix})(3+4e^{-10ix}+5e^{-100ix})\\ &= 9 + 16 + 25 + \text{cosine terms} \end{align}$$

The integral over the cosine terms is zero (why?) Therefore, your answer is $50 \cdot 2 \pi = 100 \pi$.