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Integration by Parts

Integration by Parts

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Integration by Parts

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  1. 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

  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).