4. Integration by Parts
Looking at the table of Derivative and Integral Rules, we are missing an integral version of the Product Rule. It is called Integration by Parts which is the topic of this chapter.
a1. Indefinite Integrals
In general, a derivative rule produces an indefinite integral rule: So the Product Rule produces the indefinite integral rule This integral rule is not yet in a useful form. A useful form is produced if we split up the integral on the left as the sum of two integrals and take one of them to the other side of the equation:
Integration by Parts
Notice that we have dropped the constant of integration since there is an indefinite integral on both sides which (when evaluated) will provide the constant.
This form is more useful because it helps us compute the integral of a product. If the integrand is the product of and , then we can rewrite the integral with the integrand being the product of and . Hopefully, the integral will be easier to compute.
Frequently, students incorrectly assume:
Wrong!
The integral of a product
is the product of the integrals.
Wrong!
It is NOT!
By introducing the differentials the integration by parts formula may be written in a form which is easier to remember:
Integration by Parts
where and .
Memorize this!
Notice that the integration by parts formula does not compute the integral but rather transforms it into another integral which is hopefully easier to compute. This should be clarified by the following example:
Compute
To apply the formula, we need to identify and If we take and , then and
So: Now the final integral is easy to perform:
We check by differentiating (using the Product Rule). If , then: which is the integrand we started with.
We usually write , , and in an array such as: Get in this habit.
Compute the integral .
Compute the integral .
Hint
The main difficulty is deciding what to take for and what to take for . Suppose we first try Then: This is correct but notice that the remaining integral is more complicated than the original integral. So it is better to choose the parts according to:
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