# American Institute of Mathematical Sciences

February  2022, 2(1): 73-83. doi: 10.3934/steme.2022005

## Streamlining applications of integration by parts in teaching applied calculus

 School of Engineering and Technology, Central Queensland University, Bruce Highway, North Rockhampton, QLD 4702, Australia

* Correspondence: Email: w.guo@cqu.edu.au; Tel: +61-7-49309687

Received  November 2021 Revised  January 2022 Published  March 2022

Integration by parts can be applied in various ways for obtaining solutions for different types of integrations and hence it is taught in all calculus courses in the world. However, the coverage and discourse of various applications of integration by parts in most textbooks, often packed into one section, lack a cohesion of progression for solving different types of integrals. Students may be confused by such incohesive presentation of the method and applications in the textbooks. Based on the author's experiences and practices in teaching applied calculus for undergraduate engineering and education students since 2013, a streamlined approach in teaching integration by parts has been gradually developed to the current state and ready to be shared with the mathematics teaching and learning communities. This streamlined approach allows integration by parts to be applied to solve complicated and integrated problems in a progressive way so that students can improve efficacy in their use of integration by parts gradually. This approach also makes communications easier with students on particular problems involving integration by parts.

Citation: William Guo. Streamlining applications of integration by parts in teaching applied calculus. STEM Education, 2022, 2 (1) : 73-83. doi: 10.3934/steme.2022005
##### References:
 [1] Guo, W., Unification of the common methods for solving the first-order linear ordinary differential equations. STEM Education, 2021, 1(2): 127‒140. https://doi.org/10.3934/steme.2021010 doi: 10.3934/steme.2021010. [2] Zill, D.G., A First Course in Differential Equations with Modeling Applications, 10th ed. 2013, Boston, USA: Cengage Learning. [3] Kreyszig, E., Advanced Engineering Mathematics, 10th ed. 2011, USA: Wiley. [4] Croft, A., Davison, R., Hargreaves, M. and Flint J., Engineering Mathematics, 5th ed. 2017, Harlow, UK: Pearson. [5] Stewart, J., Calculus: Concepts and Contexts, 4th ed. 2019. USA: Cengage. [6] Trim, D., Calculus for Engineers. 4th ed. 2008, Toronto, Canada: Pearson. [7] Guo, W.W., Essentials and Examples of Applied Mathematics. 2018, Melbourne, Australia: Pearson. [8] Massingham, P. and Herrington, T., Does attendance matter? An examination of student attitudes, participation, performance and attendance. Journal of University Teaching and Learning Practice, 2006, 3: 82–103. https://doi.org/10.53761/1.3.2.3 doi: 10.53761/1.3.2.3. [9] Henry, M.A., Shorter, S., Charkoudian, L., Heemstra, L.M. and Corwin, L.A., FAIL is not a four-letter word: A theoretical framework for exploring undergraduate students' approaches to academic challenge and responses to failure in STEM learning environments. CBE Life Sciences Education, 2019, 18: 1–17. https://doi.org/10.1187/cbe.18-06-0108 doi: 10.1187/cbe.18-06-0108. [10] Guo, W., Li, W. and Tisdell, C.C., Effective pedagogy of guiding undergraduate engineering students solving first-order ordinary differential equations. Mathematics, 2021, 9(14):1623. https://doi.org/10.3390/math9141623 doi: 10.3390/math9141623.

show all references

##### References:
 [1] Guo, W., Unification of the common methods for solving the first-order linear ordinary differential equations. STEM Education, 2021, 1(2): 127‒140. https://doi.org/10.3934/steme.2021010 doi: 10.3934/steme.2021010. [2] Zill, D.G., A First Course in Differential Equations with Modeling Applications, 10th ed. 2013, Boston, USA: Cengage Learning. [3] Kreyszig, E., Advanced Engineering Mathematics, 10th ed. 2011, USA: Wiley. [4] Croft, A., Davison, R., Hargreaves, M. and Flint J., Engineering Mathematics, 5th ed. 2017, Harlow, UK: Pearson. [5] Stewart, J., Calculus: Concepts and Contexts, 4th ed. 2019. USA: Cengage. [6] Trim, D., Calculus for Engineers. 4th ed. 2008, Toronto, Canada: Pearson. [7] Guo, W.W., Essentials and Examples of Applied Mathematics. 2018, Melbourne, Australia: Pearson. [8] Massingham, P. and Herrington, T., Does attendance matter? An examination of student attitudes, participation, performance and attendance. Journal of University Teaching and Learning Practice, 2006, 3: 82–103. https://doi.org/10.53761/1.3.2.3 doi: 10.53761/1.3.2.3. [9] Henry, M.A., Shorter, S., Charkoudian, L., Heemstra, L.M. and Corwin, L.A., FAIL is not a four-letter word: A theoretical framework for exploring undergraduate students' approaches to academic challenge and responses to failure in STEM learning environments. CBE Life Sciences Education, 2019, 18: 1–17. https://doi.org/10.1187/cbe.18-06-0108 doi: 10.1187/cbe.18-06-0108. [10] Guo, W., Li, W. and Tisdell, C.C., Effective pedagogy of guiding undergraduate engineering students solving first-order ordinary differential equations. Mathematics, 2021, 9(14):1623. https://doi.org/10.3390/math9141623 doi: 10.3390/math9141623.
 [1] Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131 [2] Thabet Abdeljawad. Fractional operators with boundary points dependent kernels and integration by parts. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 351-375. doi: 10.3934/dcdss.2020020 [3] Yan-An Hwang, Yu-Hsien Liao. Reduction and dynamic approach for the multi-choice Shapley value. Journal of Industrial and Management Optimization, 2013, 9 (4) : 885-892. doi: 10.3934/jimo.2013.9.885 [4] Étienne Bernard, Marie Doumic, Pierre Gabriel. Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts. Kinetic and Related Models, 2019, 12 (3) : 551-571. doi: 10.3934/krm.2019022 [5] Sebastián Ferrer, Francisco Crespo. Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction. Journal of Geometric Mechanics, 2018, 10 (3) : 359-372. doi: 10.3934/jgm.2018013 [6] Kun Li, Ting-Zhu Huang, Liang Li, Ying Zhao, Stéphane Lanteri. A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022084 [7] Fernando Hernando, Tom Høholdt, Diego Ruano. List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes. Advances in Mathematics of Communications, 2012, 6 (3) : 259-272. doi: 10.3934/amc.2012.6.259 [8] Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control and Related Fields, 2022, 12 (1) : 115-146. doi: 10.3934/mcrf.2021004 [9] Lok Ming Lui, Chengfeng Wen, Xianfeng Gu. A conformal approach for surface inpainting. Inverse Problems and Imaging, 2013, 7 (3) : 863-884. doi: 10.3934/ipi.2013.7.863 [10] Monika Muszkieta. A variational approach to edge detection. Inverse Problems and Imaging, 2016, 10 (2) : 499-517. doi: 10.3934/ipi.2016009 [11] Christopher K.R.T. Jones, Bevin Maultsby. A dynamical approach to phytoplankton blooms. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 859-878. doi: 10.3934/dcds.2017035 [12] Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757 [13] Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021, 3 (1) : 49-66. doi: 10.3934/fods.2021005 [14] Hitoshi Ishii, Paola Loreti, Maria Elisabetta Tessitore. A PDE approach to stochastic invariance. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 651-664. doi: 10.3934/dcds.2000.6.651 [15] Jorge San Martín, Takéo Takahashi, Marius Tucsnak. An optimal control approach to ciliary locomotion. Mathematical Control and Related Fields, 2016, 6 (2) : 293-334. doi: 10.3934/mcrf.2016005 [16] Antoine Gournay. A dynamical approach to von Neumann dimension. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 967-987. doi: 10.3934/dcds.2010.26.967 [17] Aihua Li. An algebraic approach to building interpolating polynomial. Conference Publications, 2005, 2005 (Special) : 597-604. doi: 10.3934/proc.2005.2005.597 [18] Elisa Gorla, Felice Manganiello, Joachim Rosenthal. An algebraic approach for decoding spread codes. Advances in Mathematics of Communications, 2012, 6 (4) : 443-466. doi: 10.3934/amc.2012.6.443 [19] Anke D. Pohl. A dynamical approach to Maass cusp forms. Journal of Modern Dynamics, 2012, 6 (4) : 563-596. doi: 10.3934/jmd.2012.6.563 [20] Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. A minimal approach to the theory of global attractors. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2079-2088. doi: 10.3934/dcds.2012.32.2079

Impact Factor: