Integration of $\cos x\cdot\cos 2x\cdot\cos 3x$ I study maths as a hobby and am trying to integrate
$\,\cos x\cdot \cos 2x\cdot \cos 3x\,$.
I am trying to split this up into a form in which I can integrate each component part. I have tried using the rule
$$2\cos A \cos B = \cos(A + B) + \cos (A - B)$$
but I can only get this far:
$$\begin{align}
\int\cos (x) \cos (2x)\cos (3x)\, dx
 & = \int\frac{1}{2}(2\cos (x)\cos (3x)\cos (2x), dx \\
 & = \int\frac{1}{2}( \cos (4x) + \cos (2x))\cos (2x), dx
\end{align}$$
but from here on I get stuck.

Use the formula twice.
$\begin{align}\cos (x)\cos (2x) \cos(3x) &= \frac12(\cos (4x) + \cos (2x)) \cos (2x) \\&= \frac12\cos(4x)\cos(2x) + \frac12 \cos(2x)\cos(2x) \\&= \frac 14 (\cos(6x) + \cos(2x)) + \frac14(\cos(4x)+1)\end{align}$
