1. Section 6.3
Integration by Parts

2. Deriving the formula
Way Back Wednesday: Write the product rule using f(x) and g(x).
As we know from other integration techniques, a derivative rule has a corresponding integration rule.
Integrate both sides of the equation in the product rule.
Rearranging the formula yields….
𝑓 𝑥 𝑔 ′ 𝑥 𝑑𝑥=𝑓 𝑥 𝑔 𝑥 − 𝑓 ′ 𝑥 𝑔 𝑥 𝑑𝑥

3. Integration by parts notation
This rule allows us to integrate some functions that we weren’t able to before. If you can rewrite the integrand as f(x) g’(x) dx, you can change the integral into something we may be able to work with.
The notation commonly replaces f(x) with u and g(x) with v, yielding….
𝑢 𝑑𝑣=𝑢𝑣 − 𝑣 𝑑𝑢
When attempting the problem, try to break up the integrand into something you can integrate (dv), and something to differentiate (u).

4. Example 1 Evaluate 𝑥 sin 𝑥 𝑑𝑥 . Can this be done using substitution?
Choose one to be u and the other to be dv.
What would happen if you chose poorly?

5. Example 2
Evaluate ln 𝑥 𝑑𝑥

6. Example 3
Sometimes an expression might require a repeated use of integration by parts.
Evaluate 𝑥 2 sin 𝑥 𝑑𝑥

7. Tabular integration
Notice how some of our examples were in the form 𝑓 𝑥 𝑔 𝑥 𝑑𝑥 , where f(x) was repeatedly differentiable to zero, and g(x) was easily integrable. When this is the case, we can take advantage of a pattern called tabular integration.
What types of functions for f(x) and g(x) would fit these criteria?
Evaluate:
𝑥 2 −5𝑥 𝑐𝑜𝑠𝑥 𝑑𝑥
𝑥 3 𝑒 𝑥 𝑑𝑥