Calculus: Techniques of Integration
Calculus: Techniques of Integration
One of very common mistake students usually do 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
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.
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.
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:
.
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
, we have two ways to go:
which gives
where you identify the two functions f(x) and g(x).
Then you need to make one derivative (of f(x)) and one integration (of g(x)) to get
.
The following examples illustrate the most common cases in
which you will be required to use Integration by Parts:
S.O.S MATHematics home page 