# Evaluating $\int\cos(7x)\cos(17x)\cos(27x)\mathrm dx$

$$\int\cos(7x)\cos(17x)\cos(27x)\mathrm dx$$ I tried using the multiple angle identities but the working out was too tedious and hairy.

• Hint: Use the product-to-sum formulae. – user137731 Jan 3 '16 at 21:10
• If you're comfortable with the fact that $\cos(x)=\frac{e^{ix}+e^{-ix}}2$, this substitution is often helpful, since you can expand the products and end up with a sum of exponential functions, which are easy to integrate. (This is equivalent to using product-to-sum identities, but not so tricky to remember) – Milo Brandt Jan 3 '16 at 21:38
• could you elaborate? @MiloBrandt – anja.wlotrzewiszczykowycki Jan 4 '16 at 7:13

$$\begin{cases} \cos(a+b) = \cos a \cos b - \sin a \sin b \\ \cos(a-b) = \cos a \cos b + \sin a \sin b \end{cases}$$
$$\cos a \cos b = \frac12 \left( \cos(a+b) + \cos(a-b) \right)$$