# Integration by parts

We can use integration by parts when we are required to integrate a product. The Edexcel formula booklet gives the formula for integration by parts:

When we integrate a product we choose one of the functions to be and the other function to be . We then find and and plug them into the formula above. It might seem like we’ve complicated things as there are two terms on the right, but the idea is to choose and so as to make the integral on the right a more simple one to find. For example, consider . We choose and as this will simplify the integral on the right hand side in the formula. It follows that and (we leave the integration constant until the end) and so:

If the integral is a definite one we can include the limits as follows:

See Example 1. One application of integration by parts may not be enough in some examples. Even though an integral might be simplified from one application, it may take another to complete the integration – see Example 2. In some of these cases, especially trigonometric products, it may seem that you end up with the same integral again. However, it is still possible to deduce the integral from this – see Example 3. Note that in this example, it doesn’t matter which way round we choose and initially.

## Derivation of the Formula

The formula above comes from the product rule for differentiation. Recall that the product rule for differentiation:

where and are both functions of . It follows that

We can rearrange this to get:

## The trick for integrating

Although we know that differentiates to , we don’t yet know how to integrate . There is a trick to this one – memorise it! Treat. as a product of one and itself. That is, write it as . We recognise this as a product and we can use integration by parts to integrate it. Let and . It follows that and . Hence,

Choosing and around the other way round would not work as choosing would require integrating – this is what we are trying to do anyway. We can check this by differentiating using the product rule where necessary:

.

When integrating other products that include it is usually best to choose as choosing it as will require integrating and will complicate the integral. See Example 1.