Integration by Parts

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.1 – Integration by Parts

Expected Educational Results

• Objective 03–01: I can use integration by parts to evaluate an indefinite integral using FTC-I.

• Objective 03–02: I can use integration by parts to evaluate a definite integral using FTC-II.

• Objective 03–03: I can use substitution and integration by parts to evaluate an indefinite integral using FTC-I.

• Objective 03–04: I can use substitution and integration by parts to evaluate a definite integral using FTC-II.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Integration by Parts

Definition: Integration by Parts

If $U=f(x)$$U=f(x)$ is a differentiable function and $V=g(x)$$V=g(x)$ is a integrable function, then $dU=f^{\prime }(x)$$dU=f^{\,\prime}(x)$ and $dV=g^{\prime }(x)$$dV=g^{\,\prime}(x)$. Thus,

$\int UdV=UV-\int VdU$$\int{U\,dV}=UV-\int{V\,dU}$

Special Cases for Integration by Parts

Investigation 09

Consider: ${\displaystyle \int \ln (x)dx}$$\displaystyle \int{\ln{(x)}\,dx}$.

1. Explain why $dV\ne \ln (x)dx$$dV\ne\ln{(x)}\,dx$.

2. Identify the appropriate expressions for $U$$U$ and $dV$$dV$.

Investigation 10

Consider: ${\displaystyle \int {\tan }^{-1}(x)dx}$$\displaystyle \int{\tan^{-1}{(x)}\,dx}$.

1. Explain why $dV\ne {\tan }^{-1}(x)dx$$dV\ne\tan^{-1}{(x)}\,dx$.

2. Identify the appropriate expressions for $U$$U$ and $dV$$dV$.

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Created: Tuesday, 16 August 2022 - 16:58 (EDT)