Energy of a signal $s(t)=\int_{-4}^{3} e^{{\pi}{i}{t}{u}} du$

Question

I need to calculate the energy of a signal: $s(t)=\int_{-4}^{3} e^{{\pi}{i}{t}{u}} du$. I know the energy can be calculated from $<s,s>=\int_{-\infty}^{\infty} (s(t)*\overline{s(t)})dt$. The formula for $s$ gives $s=\frac{1}{{i}{\pi}{t}}(e^{3{\pi}it}-e^{-4{\pi}it})$ From this the energy $E=\int_{-\infty}^{\infty}\frac{-1}{{\pi}^2{t^2}}(2-e^{{\pi}it7}-e^{-{\pi}it7}) dt$. This is far as i can get.

Answer 1

An idea (perhaps with complex integration, but perhaps you already know it):

$$2-e^{7\pi i t}-e^{-7\pi it}=-\left(e^{\frac72\pi it}-e^{-\frac72\pi it}\right)^2=4\sin^2\frac72\pi t\implies$$

$$\frac4{\pi^2}\int_{-\infty}^\infty\frac{\sin^2\frac72\pi t}{t^2}=\frac4{\pi^2}\frac{7\pi^2}2=14$$