Verify the following trigonometric integral for integer (k) : [ int_{0}^{2 pi} sin (x / 2) cos

Question:

Verify the following trigonometric integral for integer $k$ :

\[
\int_{0}^{2 \pi} \sin (x / 2) \cos (k x) d x=\frac{-4}{4 k^{2}-1} .
\]

Hence find the coefficients $\lambda_{k}$ in the Fourier expansion of the function

\[
2 / \pi-\sin (x / 2)=\sum_{k=0}^{\infty} \lambda_{k} \cos (k x)
\]

for $0 \leq x<2 \pi$, and show that they are all positive.