Let $f\in L^1(\mathbb{T})$ where $\mathbb{T}$ is the unit circle in the complex plane. I need to calculate $\frac{1}{2\pi}\int_{-\pi}^{\pi}\left(\int_{0}^{t}f(e^{i \tau})d\tau\right) e^{(-int)}dt$. The answer is $\frac{\hat{f}(n)}{in}$. Is my following approach correct?
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left(\int_{0}^{t}f(e^{i \tau})d\tau\right) e^{(-int)}dt$
=$\frac{1}{2\pi}\int_{-\pi}^{\pi}f(e^{it})\left(\int_{0}^{t} e^{(-in \tau)}d\tau\right)dt$ (if we substitute $\tau=t$)
=$\frac{\hat{f}(n)}{in}$