Energy of 5cosπt + sin5πt

$x(t) = 5\cos\pi t + \sin 5\pi t$ exists for all time and does not decay, so its energy is expected to be infinite. Check with the definition.

Periods: $\cos\pi t$ has $T_1 = 2$ s, $\sin 5\pi t$ has $T_2 = 2/5$ s; the ratio is rational, so $x(t)$ is periodic with $T_0 = \text{LCM}(2, 0.4) = 2$ s.

$$x^2(t) = 25\cos^2\pi t + \sin^2 5\pi t + 10\cos\pi t\,\sin 5\pi t$$

Energy in one period:

$$\begin{aligned} E_{T_0} &= \int_0^{2}25\cos^2\pi t\,dt + \int_0^{2}\sin^2 5\pi t\,dt + 10\int_0^{2}\cos\pi t\sin5\pi t\,dt \\ &= 25(1) + 1 + 0 = 26 \end{aligned}$$

(The cross term is zero because sinusoids of different frequencies are orthogonal over a common period.)

Total energy over $-\infty < t < \infty$ = (number of periods) × 26:

$$E = \lim_{k\to\infty} k \times 26 = \infty$$

Average power:

$$P = \frac{E_{T_0}}{T_0} = \frac{26}{2} = 13 \quad\left(= \frac{5^2}{2} + \frac{1^2}{2}\right)$$

Answer: $E = \infty$; the signal is a power signal with $P = 13$ W (normalised).