Question

find the integral :
cos3xcosx dx
thanks.

Answer 1

Use the following identities:
\[\cos(a+b)x=\cos ax\cos bx-\sin ax \sin bx\\
\cos(a-b)x=\cos ax\cos bx+\sin ax\sin bx\]
Add the two together, you have
\[\cos(a+b)x+\cos(a-b)x=2\cos ax\cos bx\\
\cos ax\cos bx=\frac{1}{2}(\cos(a+b)x+\cos(a-b)x)\]
where \(a=3\) and \(b=1\).