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}$

Using Integration by Parts with a Substitution

Investigation 06

Explain why $u$$u$-substitution IS a valid first step for the following integrals:

1. ${\displaystyle \int \sin \left(\sqrt{x}\right)dx}$$\displaystyle\int{\sin{\left(\sqrt{x}\right)}\,dx}$

2. ${\displaystyle \int x^3{e}^{x^2}dx}$$\displaystyle\int{x^3e^{x^2}\,dx}$

3. ${\displaystyle \int e^{\sqrt{x}}dx}$$\displaystyle \int{e^{\sqrt{x}}\,dx}$

4. ${\displaystyle \int \sin (e^x)dx}$$\displaystyle \int{\sin{(e^x)}\,dx}$

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Tuesday, 16 August 2022 - 15:07 (EDT)