Product-to-sum and double-angle identities

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Convert products with different frequencies into sums that are easy to integrate. Adjust for each frequency when finding primitives.

Step-by-step example

1. Convert the product to a sum.

   $$\sin(2x)\cos(3x)=\tfrac12[\sin(5x)-\sin x]$$

2. Integrate each sine.

   $$\int\sin(kx)\,dx=-\cos(kx)/k$$

3. Combine the primitives.

   $$I=-\frac1{10}\cos(5x)+\frac12\cos x+C$$