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By Rick Stoll          

One of very common mistake people new to intigration is

To convince yourself that it is a wrong formula, take f(x) = x and g(x)=1. Therefore, one may wonder what to do in this case. A partial answer is given by what is called Integration by Parts. In order to understand this technique, recall the formula

which implies

Therefore if one of the two integrals and is easy to evaluate, we can use it to get the other one. This is the main idea behind Integration by Parts. Let us give the practical steps how to perform this technique:

1

Write the given integral

where you identify the two functions f(x) and g(x). Note that if you are given only one function, then set the second one to be the constant function g(x)=1.

2

Introduce the intermediary functions u(x) and v(x) as:

Then you need to make one derivative (of f(x)) and one integration (of g(x)) to get

Note that at this step, you have the choice whether to differentiate f(x) or g(x). We will discuss this in little more details later.

3

Use the formula

 

4

Take care of the new integral .

 

The first problem one faces when dealing with this technique is the choice that we encountered in Step 2. There is no general rule to follow. It is truly a matter of experience. But we do suggest not to waist time thinking about the best choice, just go for any choice and do the calculations. In order to appreciate whether your choice was the best one, go to Step 3: if the new integral (you will be handling) is easier than the initial one, then your choice was a good one, otherwise go back to Step 2 and make the switch. It is after many integrals that you will start to have a feeling for the right choice.

In the above discussion, we only considered indefinite integrals. For the definite integral , we have two ways to go:

1

Evaluate the indefinite integral

which gives

 

2

Use the above steps describing Integration by Parts directly on the given definite integral. This is how it goes:

(i)

Write down the given definite integral

where you identify the two functions f(x) and g(x).

(ii)

Introduce the intermediary functions u(x) and v(x) as:

Then you need to make one derivative (of f(x)) and one integration (of g(x)) to get

 

(iii)

Use the formula

 

(iv)

Take care of the new integral .

 

The following examples illustrate the most common cases in which you will be required to use Integration by Parts:

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Last modified: April 30, 2001