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

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

Investigation 01 – Reprise

1. What do you notice about the integrand on the left side of the equation in the definition above? Explain.

2. What do you notice about the right side of the equation in the definition above? Explain.

Check your Understanding

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Last Modified: Tuesday, 10 August 2021 12:05 EDT