${\int }_0^{\mathrm{\pi }} \cos 6\mathrm{x}\cdot {\cos }^8\mathrm{xdx}=$

$$\begin{gathered} {\cos }^8\mathrm{x}=\frac{1}{2^8}{\left({\mathrm{e}}^{ix}+{\mathrm{e}}^{-ix}\right)}^8=\frac{1}{2^8}\left(({\mathrm{e}}^{i8\mathrm{x}}+e^{-i8x})+8({\mathrm{e}}^{6\mathrm{ix}}+e^{-i6x})+28(e^{i4x}+e^{-i4x})+56(e^{i2x}+e^{-i2x})+70\right) \\ =\frac{1}{2^7}(\cos 8x+8\cos 6x+28\cos 4x+56\cos 2x+35) \end{gathered}$$
Using binomial Expansion  $\left.(\mathrm{x}+\mathrm{y}{)}^{\mathrm{n}}{=}^{\mathrm{n}}{\mathrm{C}}_0{\mathrm{x}}^{\mathrm{n}}{+}^{\mathrm{n}}{\mathrm{C}}_1{\mathrm{x}}^{\mathrm{n}-1}{\mathrm{y}}^1+\dots ..\right]$
$\begin{array}{l}=\frac{1}{2^7}(\cos 8\mathrm{x}+8\cos 6\mathrm{x}+\dots \dots )\\ \begin{array}{rl} & {\int }_0^{\mathrm{\pi }} \cos 6\mathrm{x}\cdot {\cos }^8\mathrm{xdx}={\int }_0^{\mathrm{\pi }} \cos 6\mathrm{x}\cdot \frac{8}{2}\cos 6\mathrm{xdx}\\ & =\frac{1}{16}{\int }_0^{\mathrm{\pi }} {\cos }^{2}6\mathrm{xdx}={\int }_0^{\mathrm{\pi }} \frac{1+\cos 12\mathrm{x}}{32}\mathrm{dx}=\frac{\mathrm{\pi }}{32}\end{array}\end{array}$