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(****) (a) Find the indefinite integrals below, noting that the expression $e^{-2x} (A\cos(mx) + B\sin(mx))$ might be relevant:
\begin{align*}
\int e^{-2x} \sin(mx) dx &= \\
\int e^{-2x} \cos(mx) dx &= \\
\end{align*}
(b) A random variable $Y$ has an exponential distribution, with the probability density shown here:
$$p(y) = \begin{cases} 2e^{-2y}, & \text{for } y \ge 0, \\ 0, & \text{for } y < 0. \end{cases}$$
Find the expected value of the random variable $X = 6 \cos(6Y) + 6 \sin(6Y)$.
$E[X] = E[6 \cos(6Y) + 6 \sin(6Y)] = $
Hint: For any given random variable $Y$ and function $h$, $E[h(Y)] = \int_{-\infty}^{\infty} h(y)p(y) dy$.
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