Sometimes we will not be able to use a substitution to help us integrate a complicated function. If the integral is a product we can use an alternative method.

Integration by parts is the inverse of the product rule. Integrating the product rule with respect to x derives the formula:

sometimes shown as

To integrate a product (that cannot be easily multiplied together), we choose one of the multiples to represent u and then use its derivative, and choose the other multiple as dv/dx and use its integral.

Example:

To integrate

we let u = x and dv/dx=e^2x

This gives us:

Putting these into the formula we get:

Note: Most questions that require integration by parts will have x or x² as one of the multiples. Substitute this with u, and let dv/dx be the other multiple.

If one multiple is an x² then we will need to use integration by parts twice.

When using limits apply the limits to all of the integration.

ln x

If one of the multiples is ln x then this will have to be substituted with u as we can easily differentiate this using:

This enables us to find

Example:

To integrate

let u = ln x and dv/dx = 1.

Put these into the formula to get:

sin x and cos x

sin x and cos x are functions that follow a pattern through differentiation. After differentiating twice you are effectively back where you started. This idea can be used to integrate seemingly impossible expressions.

Example:

Find      

Therefore:

(Now use parts again.)

Therefore:

Putting these two together we get that:

This integral is in terms of the original question! It can be rearranged to give,

Therefore:

Note: In an examination you will get lots of guidance from the question if you have to do an integration like this!