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## Integration by Parts

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**Integration by Parts**• Method of Substitution • 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.**Helpful Hints**• For u, choose a function whose derivative is “nicer”. • LIATE • dv must include everything else (including dx).