1. Integration by Parts Method of Substitution Integration by Parts
Related to the chain rule
Integration by Parts
Related to the product rule
More complex to implement than the Method of Substitution

2. Derivation of Integration by Parts Formula
Let u and v be differentiable functions of x.

3. Derivation of Integration by Parts Formula
Let u and v be differentiable functions of x.
(Product Rule)

4. Derivation of Integration by Parts Formula
Let u and v be differentiable functions of x.
(Product Rule)
(Integrate both sides)

5. Derivation of Integration by Parts Formula
Let u and v be differentiable functions of x.
(Product Rule)
(Integrate both sides)
(FTC; sum rule)

6. Derivation of Integration by Parts Formula
Let u and v be differentiable functions of x.
(Product Rule)
(Integrate both sides)
(FTC; sum rule)

7. Derivation of Integration by Parts Formula
Let u and v be differentiable functions of x.
(Product Rule)
(Integrate both sides)
(FTC; sum rule)
(Rearrange terms)

8. Integration by Parts Formula
What good does it do us?
We can trade one integral for another.
This is only helpful if the integral we start with is difficult and we can trade it for a good (i.e., solvable) one.

9. Classic Example

10. Helpful Hints For u, choose a function whose derivative is “nicer”.
LIATE
dv must include everything else (including dx).