# 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:

and .

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.

## Examples

Integrate .

Solution:

Recall the double angle formula . This can be rewritten as . It follows that . Hence,

Find the exact value of .

Solution:

Firstly, expanding the brackets gives:

where we transformed terms into those that we can integrate. See Example 1 for more on transforming . Hence,

Find .

Solution:

Using the formula we can write . It follows that we must find and such that and or and . So, and . Hence, . Note that cos is an even function (see more on odd/even functions) and so . The integral becomes:

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