Consider the definite integral $\displaystyle D=\int_{-\pi}^{\pi}\cos\lambda x\cos nx\,dx$ where n is a positive integer and lamda is a positive real number.

Given also that $\displaystyle D=\left\{\begin{matrix}

\pi & $for$\,\lambda =n\\

0 & $for$\,\lambda \, $a positive integer$\,,\lambda \neq n\\

\frac{(-1)^n2\lambda \sin(\lambda \pi )}{\lambda ^2-n^2} & $for$\,\lambda \,$not an integer$

\end{matrix}\right.$

Show that when $\displaystyle 0< \lambda < n,\,|D|< \pi$ and when $\displaystyle \lambda > n+\frac{1}{2},\,|D|< 3$

Help of any kind would be greatly appreciated.