Integration by Parts
Integration by Parts (IBP) is a special method for integrating products of functions. For example, the following integrals
in which the integrand is the product of two functions can be solved using integration by parts.
This method is based on the product rule for differentiation.
Suppose that u (x) and v (x) are differentiable functions. Then the product rule in terms of differentials gives us:
Rearranging this rule, we can write
Integrating both sides with respect to x results in
This is the integration by parts formula. The goal when using this formula is to replace one integral (on the left) with another (on the right), which can be easier to evaluate.
The key thing in integration by parts is to choose u and dv correctly.
The acronym ILATE is good for picking u. ILATE stands for
The closer a function is to the top, the more likely that it should be used as
Solved Problems
Example 1.
Compute
Solution.
We use integration by parts:
Let
Hence, the integral is
Example 2.
Integrate
Solution.
We can choose
so we can easily integrate it and find the function
Apply the integration by parts formula:
The last integral is well known:
Hence
Example 3.
Evaluate the integral
Solution.
We choose
Hence
Substituting these expressions into the integration by parts formula
we have
Example 4.
Integrate
Solution.
We are to integrate by parts:
Example 5.
Evaluate the integral
Solution.
Using the ILATE rule, we can choose
Then
Integrating by parts, we obtain
Example 6.
Evaluate the integral
Solution.
To use integration by parts we rewrite the integral as follows:
Now we can apply the ILATE rule, that is
This yields
Integrating by parts, we have
Example 7.
Compute the integral
Solution.
Keeping in mind the ILATE rule, we can choose
Then
This yields:
Example 8.
Evaluate the integral
Solution.
We set
Then
Using the integration by parts formula
we have