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.