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Integration By Parts

Objectives

  • State the integration by parts formula.
  • Use the method of integration by parts to integrate products of certain polynomial, trigonometric, and exponential functions.
  • Use the method of integration by parts to integrate logarithmic and inverse trigonometric functions.

Summary

The integration by parts formula is an integration version of the product rule for derivatives. From the product rules for derivatives ( f ( x ) g ( x ) ) ′ = f ′ ( x ) g ( x ) − f ( x ) g ′ ( x ) we integrate both sides and arrange terms to find that

∫ f ( x ) g ′ ( x ) d x = f ( x ) g ( x ) − ∫ f ′ ( x ) g ( x ) d x

Another version of the integration by parts formula is found by substituting u = f ( x ) and v = g ( x ) into this last equation to get:

The purpose of the integration by parts formula is to change the integral of a product f ( x ) g ′ ( x ) into an integral of the product f ′ ( x ) g ( x ) . To use the integration by parts formula,

  1. Identify one part of the integrand as u = f ( x ) . The remaining portion is identified as d v = g ′ ( x ) d x .
  2. With the choice of u and d v , compute the differential d u = f ′ ( x ) d x and integrate to find v = g ( x ) .
  3. Substitute u , v , d u into u v − ∫ v d u .

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