Integration by Parts Method of Substitution Integration by Parts

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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

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

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

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

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

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

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)

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.

Classic Example

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