Calculus: Techniques of Integration

Integration by Parts

One of very common mistake students usually do is

displaymath178

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

displaymath184

which implies

displaymath186

Therefore if one of the two integrals tex2html_wrap_inline188 and tex2html_wrap_inline190 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

displaymath192

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:

displaymath204

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

displaymath210

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

displaymath216

4
Take care of the new integral tex2html_wrap_inline218 .

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 waste 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 tex2html_wrap_inline220 , we have two ways to go:

1
Evaluate the indefinite integral

displaymath222

which gives

displaymath224

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

displaymath226

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

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

displaymath204

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

displaymath210

(iii)
Use the formula

displaymath244

(iv)
Take care of the new integral tex2html_wrap_inline246 .

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


[Calculus]
[Geometry] [Algebra] [Trigonometry ]
[Differential Equations] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi

Copyright 1999-2014 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour