On this page, I provide examples of U-Substitution, Integration By Parts and Trigonometric Substitution. I do not normally provide examples, but integration requires a great deal of time to recognize the various types and the methods to solve them. This is a good starting place for your edification, not for the fainthearted and remember MATH IS FUN. Also, don’t forget to understand the Fundamental Theorem of Calculus!
Here are some integration rules and cheat sheets to assist you when learning and performing integral calculus.
| Common Functions | Function | Integral |
| Constant | ∫a dx | ax + C |
| Variable | ∫x dx | x2/2 + C |
| Square | ∫x2 dx | x3/3 + C |
| Reciprocal | ∫(1/x) dx | ln|x| + C |
| Exponential | ∫ex dx | ex + C |
| ∫ax dx | ax/ln(a) + C | |
| ∫ln(x) dx | x ln(x) − x + C | |
| Trigonometry (x in radians) | ∫cos(x) dx | sin(x) + C |
| ∫sin(x) dx | -cos(x) + C | |
| ∫sec2(x) dx | tan(x) + C | |
| Rules | Function | Integral |
| Multiplication by constant | ∫cf(x) dx | c∫f(x) dx |
| Power Rule (n≠−1) | ∫xn dx | |
| Sum Rule | ∫(f + g) dx | ∫f dx + ∫g dx |
| Difference Rule | ∫(f – g) dx | ∫f dx – ∫g dx |
| Integration by Parts | See Integration by Parts | |
| Substitution Rule | See Integration by Substitution |
“Calculus Cheat Sheet”. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/pdf/Calculus_Cheat_Sheet_All.pdf.
⭐ Ryan, Mark. “Calculus Workbook For Dummies Cheat Sheet – Dummies”. 2022. dummies.com. https://www.dummies.com/article/academics-the-arts/math/calculus/calculus-workbook-for-dummies-cheat-sheet-208666/.
U-Substitution
As I was taught, U-Substitution is a way of dealing with the chain rule from differentiation: It reverses it! The chain rule deals with derivatives of composite functions. In examples like this, we say that the derivative of the function f(g(x)) is f’(g(x))*g’(x). This is why the derivative of -cos(2x) isn’t just sin(2x): we are missing an extra factor of 2 from the derivative of the inside function 2x. Now it should be apparent to us why integrating sin(2x) doesn’t simply yield -cos(2x). It is absolutely necessary to “account” for the chain rule in both differentiation and integration problems. Let’s look at an example problem together.1
See the following web pages:
“Integration By Substitution – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Integration_by_substitution.
“Integration Using Substitution Method (Solved Problems)”. 2022. BYJUS. https://byjus.com/maths/integration-substitution/.
“U-Substitution”. 2022. math.ucdavis.edu. https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/usubdirectory/USubstitution.html.
⭐ “𝘶-Substitution Intro (Video) | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-9/v/u-substitution.
Integration by Parts
Integrating by parts is the integration version of the product rule for differentiation. The basic idea of integration by parts is to transform an integral you can’t do into a simple product minus an integral you can do. Here’s the formula:3
Don’t try to understand this yet. Wait for the examples that follow.
If you remember that, you can easily remember that the integral on the right is just like the one on the left, except with the u and v reversed.
See the following web pages:
“How To Do Integration By Parts – Dummies”. 2022. dummies.com. https://www.dummies.com/article/academics-the-arts/math/calculus/how-to-do-integration-by-parts-192235/.
“Integration By Parts – Formula, ILATE Rule & Solved Examples”. 2022. BYJUS. https://byjus.com/maths/integration-by-parts/.
“Integration By Parts – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Integration_by_parts.
⭐ “Integration By Parts Intro (Video) | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/ap-calculus-bc/bc-integration-new/bc-6-11/v/deriving-integration-by-parts-formula.
Trigonometric Substitution
This method works when the integrand contains radicals of the forms2
(or powers of these roots), where a is a constant and u is an expression in x.
See the following web pages:
“Calculus II – Trig Substitutions”. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx.
“How Do You Integrate sin(x)cos(x)? – Maths Q&A”. 2022. BYJU’s. https://byjus.com/questions/how-do-you-integrate-sin-x-cos-x/.
“Integration by Trigonometric Substitution – Calculus | Socratic”. 2022. socratic.org. https://socratic.org/calculus/techniques-of-integration/integration-by-trigonometric-substitution.
“Integration of Tan X – Formula, Derivation And Examples”. 2022. BYJUS. https://byjus.com/maths/integration-of-tan-x/.
⭐ “Introduction To Trigonometric Substitution (Video) | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/integral-calculus/ic-integration/ic-trig-substitution/v/introduction-to-trigonometric-substitution.
“Trigonometric Substitution – A Tool for Evaluating Integrals”. 2022. web.iit.edu. https://web.iit.edu/sites/web/files/departments/academic-affairs/academic-resource-center/pdfs/Trigonometric_Substitution.pdf.
“Trigonometric Substitution – Calculus Tutorials”. 2022. math.hmc.edu. https://math.hmc.edu/calculus/hmc-mathematics-calculus-online-tutorials/single-variable-calculus/trigonometric-substitution/.
“Trigonometric Substitution – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Trigonometric_substitution.
References
1 Ming, Albert. “3 Important Methods Of Integration To Know”. 2021. Medium. https://albertming88.medium.com/3-important-methods-of-integration-to-know-dfc6108f8eea.
2 “Calculus Workbook For Dummies Cheat Sheet – Dummies”. 2022. dummies.com. https://www.dummies.com/article/academics-the-arts/math/calculus/calculus-workbook-for-dummies-cheat-sheet-208666/.
3 “How To Do Integration By Parts – Dummies”. 2022. dummies.com. https://www.dummies.com/article/academics-the-arts/math/calculus/how-to-do-integration-by-parts-192235/.
Additional Reading
⭐ Ryan, Mark. Calculus for Dummies. Indianapolis, Indiana: Wiley Publishing, Inc., 2003.
“Integration Rules”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/integration-rules.html.
“Integration By Parts”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/integration-by-parts.html.
“Integration By Substitution”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/integration-by-substitution.html.
“Introduction To Integration”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/integration-introduction.html.
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
Mathematical Mysteries 