Chapter 7 Additional Integration Topics. Section 3 Integration by Parts. Learning Objectives for Section 7.3 Integration by Parts. The student will be able to integrate by parts. An Unsolved Problem. In a previous section we said we would return later to the indefinite integral.
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Chapter 7Additional Integration Topics
Integration by Parts
The student will be able to integrate by parts.
In a previous section we said we would return later to the indefinite integral
since none of the integration techniques considered up to that time could be used to find an antiderivative for ln x. We now develop a very useful technique, called integration by parts that will enable us to find not only the preceding integral, but also many others.
Integration by parts is based on the product formula for derivatives:
We rearrange the above equation to get:
Integrating both sides yields
which reduces to
If we let u = f (x) and v = g(x) in the preceding formula, the equation transforms into a more convenient form:
Integration by Parts Formula
This method is useful when the integral on the left side is difficult and changing it into the integral on the right side makes it easier.
Remember, we can easily check our results by differentiating our answers to get the original integrals.
Let’s look at an example.
Our previous method of substitution does not work. Examining the left side of the integration by parts formula yields two possibilities.
Let’s try option 1.
We have decided to use u = x and dv = e x dx.
This means du = dx and v = e x, yielding
This is easy to check by differentiating.
Note: Option 2 does not work in this example.
Let u = ln x and dv = x3 dx, yielding du = 1/xdx and v = x4/4
Check by differentiating:
Recall that our original problem was to find
We follow the suggestion
For integrals involving x p(ln x)q, try u = (ln x)q and dv = x p dx
Set p = 0 and q = 1 and choose u = ln x and dv = dx.
Then du = 1/xdx and v = x, hence
We have developed integration by parts as a useful technique for integrating more complicated functions.
Integration by Parts Formula: