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I am mostly confused on how to manipulate the integrand. I know that I will likely have to use Euler formula or hyperbolic cosine, but I am very confused as to what where I must begin. Thank you.
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This is a pretty common integral for EE, its used for finding signal power. As you mentioned, Eulers will be particularly helpful. Start off by squaring the integrand. From there, use Eulers and see where that leads.
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cos(x) = (e^(ix)+e^(-ix))/2

The answer will depend upon α and ω0. Unless they're defined in terms of T.
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