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}$

The Need of Integration by Parts

Investigation 03

Explain why $u$$u$-substitution is NOT a valid method for the following integrals:

1. ${\displaystyle \int x{e}^{x}dx}$$\displaystyle\int{xe^x\,dx}$​

2. ${\displaystyle \int x\sin (x)dx}$$\displaystyle\int{x\sin{(x)}\,dx}$

3. ${\displaystyle \int x^2\cos \left(2x\right)dx}$$\displaystyle\int{x^2\cos{\left(2x\right)}\,dx}$

4. ${\displaystyle \int x^3{e}^{6x}dx}$$\displaystyle\int{x^3e^{6x}\,dx}$

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

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Last Modified: Tuesday, 10 August 2021 15:19 EDT