Integration by parts

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Integration by parts

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:

Formula

sometimes shown as

Alternative formula

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

Example

we let u = x and dv/dx=e2x

This gives us:

Example

Putting these into the formula we get:

Example

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:

Derivative of In x

This enables us to find

Example

Example:

To integrate

Example

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

Example

Put these into the formula to get:

Example

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      Example

Example

Therefore:

Example

(Now use parts again.)

Example

Therefore:

Example

Putting these two together we get that:

Example

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

Example

Therefore:

Example

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

S-cool exclusive!!