Integration Using Double Angle Formulae
In order to integrate , for example, it might be tempting to use the basic trigonometric identity as this identity is more familiar. However, integrating is more complicated than integrating itself. Instead, we can use a double angle identity to integrate . Recall the double angle formulae:
The final identity for can be rearranged to get an expression for . That is, . This is easy to integrate and so:
A similar process can be applied to integrate . We can also integrate when the argument is different – see Example 1. To integrate , we use a reciprocal trigonometric identity. See below.
Using Reciprocal Trigonometric Identities
We can use the double angle formulae to integrate and . However, for integrating we may use a reciprocal trigonometric identity. Recall that . Hence,
using the formula booklet to integrate . See Example 2.
Using Compound Angle Formulae
It is important to remember, as well as the above, that a question may ask you to integrate a trigonometric function which, at first, looks hugely unfamiliar. For example, how could we integrate ? It might be tempting to try integration by parts since it is a product. However, the formula booklet provides compound angle identities that will prove useful in integrating this kind of function:
The first formula will help us to integrate . We rewrite the formula as . The individual terms on the right of this will be much easier to integrate than the product on the left. Hence, we must find and such that and . It follows that and . Hence, and . The integral becomes:
Note that or could be negative and we would have to use the odd/even properties of sin/cos – see Example 3. It is possible to integrate using integration by parts but it is much simpler to use the method above. See Integration by Parts Example 3.