Solving the integral $\int_{0}^{\pi/2}(\sin[2n+1]t) \cdot \frac{\sin(t)-t}{sin(t) \cdot t}$

Question

How should I solve this integral ?:

$$ \int_{0}^{\pi/2}\sin\left(\vphantom{\Large A}\left[2n + 1\right]t\right)\, {\sin\left(t\right) - t \over t\sin\left(t\right)}\,{\rm d}t $$ How can I use Fourier coefficients ?.

Answer 1

$\int_{0}^{\pi/2}\sin\bigg([2n+1]t\bigg)\cdot \Big(\frac{1}{t}-\frac{1}{\sin(t)}\Big) \mathrm{d}t$

I noticed that function $\frac{1}{t}-\frac{1}{\sin(t)}$ is almost everywhere continuous so I used Riemann's Lema. The answer is $0$. Is it right?