Integration by Parts Formula
Unlike derivatives, there is no formula that can integrate any product of functions.
The closest thing we have to a rule for integrating products of functions is integration by parts. Interestingly, it is based on the formula for differentiating a product of functions.
However, integration by parts transforms an integral of a product into another integral. This formula does not work for integrating all products of functions.
The integration by parts formula is:
Notice that we have to differentiate 
and integrate 
, so it will be convenient for the integral of to be simple.
In general, polynomial, logarithmic, and arctangent functions are chosen as . Meanwhile, exponential, sine, and cosine functions are chosen as .
Derivation of the Formula
Suppose we have the functions 
and 
. Then their derivative is given by:
If we integrate both sides of the equation, we obtain:
Then, if we move 
to the left side, we obtain:
which is the formula we were looking for.
Proposed Exercises
We have a product between the function 
and 
. As mentioned previously, in this type of case we choose 
and 
.
We differentiate :
We integrate 
:
So the integral becomes:
Thus,
We have a product between the function 
and . In this type of case we choose and 
.
We differentiate :
We integrate :
So the integral becomes:
Thus,
 
We have a product between the function 
and 
. In general, both functions are usually taken as ; however, in this type of case the logarithm takes preference and we choose 
and 
.
We differentiate (this is the reason we chose the logarithm):
We integrate :
So the integral becomes:
Thus,
We have a product between the function and 
. Again, in this type of case we choose and 
(the logarithm function is always chosen as ).
We differentiate :
We integrate :
So the integral becomes:
Thus,
Summarize with AI:
Did you like this article? Rate it!
Loading...
