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Integration By Parts

Suppose we have two functions multiplied by each other

and differentiate according to the product rule:

then by integrating both sides between the limits a and b

and rearranging gives

 

or as the formula is better known

INTEGRATION BY PARTS

 

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Tricks: If one of the functions is a polynomial (say nth order) and the other is integrable n times, then you can use the fast and easy Tabular Method:

Tabular Method

Suppose and . Then if we set up a table, differentiating f(x) as many times as it takes to get to zero and integrating g(x) as many times, we get

- notice how we alternate the signs in the “I” column. Then, multiplying rows with like letters (which you can skip on your own paper to help make this clearer - draw arrows instead from (a) to (a), etc.) gives the following directly as the anti-derivative:

This method is much faster than the f-g method or the older u-v, especially for iterated (more than once) integrals by parts (Thanks to Dr. William T. Guy, UT Austin).

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Advanced

There is a way to extend the tabular method to handle arbitrarily large integrals by parts - you just include the integral of the product of the functions in the last row and pop in an extra sign (whatever is next in the alternating series), so that

The trick is to know when to stop for the integral you are trying to do. Try it for a few simple functions, you'll see!

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The u - v Method:

This is the old standby

which is identical to the canonical method if you set

Many find this a lot easier to remember!

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Examples || 1 || 2 || 3 ||

Example 1

D	I

So according to the table,

With the u - v method,

and

up

Example 2

D	I
	x

So according to the table,

up

Example 3

D	I

This one is a bit more fun… from the table we have

The last integral on the right yields to the substitution

and , viz.

which makes the final integral

up

So this method can be used for even very nasty integrals!

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Exercises || 1 || 2 || 3 ||

These are really proofs since I give you the answers.

Use the above methods to show

1.

2.

3.

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Recommended Books

  Schaum's Outline of Calculus (Schaum's...

    The classic calculus problem book - very light on theory, plenty of problems with full solutions, more problems with answers

 

 

  Schaum's Easy Outline: Calculus

    A simplified and updated version of the classic Schaum's Outline. Not as complete as the previous book, but enough for most students

 

 

 

 

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Keywords

integral, integration, technique, parts, products, tabular, u-v

-please report any errors immediately!

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