why $\int_{-\pi}^{\pi}f_n(t)^p$ is not zero on $L^p[-\pi,\pi]$?

Question

$$f_n(t)=e^{int}$$ I have to calculate this $\int_{-\pi}^{\pi} e^{intp}\,dt$. Can i use dominated convergence theorem?

Answer 1

Since the integrand has magnitude $1$ and the interval is finite, dominant convergence is trivially irrelevant. The integral is obvious.

$e^{intp}=cos(ntp)+isin(ntp)$. Since $sin(x)$ is odd, the integral=$\frac{2sin(\pi np)}{np}$, which $=0$ for even integer $np$ and $\ne 0$ otherwise..