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Integration by Parts

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 .

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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).

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