Calculus textbooks with good sections on integration

In summary, the conversation discusses the topic of integration, specifically integration by substitution, and the need for recommendations on books with good treatment of different techniques of integration. Some suggestions are given, including "Inside Interesting Integrals" by Nahin, "Irresistible Integrals" by George Boros, and online resources such as OpenStax and Physics Forums. The conversation also touches on the relationship between integration and the chain rule and product rule of differentiation, as well as the use of identities and formulas in solving integrals. The speaker mentions their background in studying calculus and the need for additional resources to improve their understanding.
  • #1
Santiago24
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Hi I'm having troubles with integration specially by substitution, I'm going to read a calculus textbook and i need recommendations of books with a good treatment on the different techniques of integration. I'd like a book with good exercises for self study and a exposure to integration of different functions like integrals involving logarithmic and trigonometric functions or integrals involving logarithmic and power functions.
 
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  • #2
For one specifically on integration I guess you could try 'inside interesting integrals' by Nahin? My old maths teacher used to rave about that one a lot, although I didn't read the whole thing yet.
 
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  • #4
You might have already realized this for yourself but in case you haven’t I’ll say it:

1) U-substitution is related to the chain rule of differentiation

2) UV-substitution (or integration by parts) is related to the product rule of differentiation

For UV-substitution there is a general rule of thumb for which function you want to make equal to ##u## in that order.

Log, Inverse Trig, Algebraic, Trig, Exponential.

1) and 2) might be regarded as trivial by some but if you don’t know them calculus becomes WAY more difficult.
 
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  • #5
General Kenobi said:
For one specifically on integration I guess you could try 'inside interesting integrals' by Nahin? My old maths teacher used to rave about that one a lot, although I didn't read the whole thing yet.
I was looking on the internet and a lot of people recommend it and other book named "Irresistible Integrals" by George Boros, so i'll check it thanks.
 
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  • #7
PhDeezNutz said:
You might have already realized this for yourself but in case you haven’t I’ll say it:

1) U-substitution is related to the chain rule of differentiation

2) UV-substitution (or integration by parts) is related to the product rule of differentiation

For UV-substitution there is a general rule of thumb for which function you want to make equal to ##u## in that order.

Log, Inverse Trig, Algebraic, Trig, Exponential.

1) and 2) might be regarded as trivial by some but if you don’t know them calculus becomes WAY more difficult.
Hi thanks for the information, my problem with the integration of trigonometric functions is that they use a lot of identities or formulas that i don't know where they come.
 
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  • #8
What is your calculus background? You seem to be gravitating towards “how to solve hard integrals” but your statements seem more “how to address these weaknesses”
 
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  • #9
caz said:
What is your calculus background? You seem to be gravitating towards “how to solve hard integrals” but your statements seem more “how to address these weaknesses”
Hi I'm self studying calculus with the Spivak book. Yes, i have problems with integration by substitution because when i have to replace for u i don't know how to make appear du.
 
  • #10
If u=f(x), du=f’(x)dx

You need to go through a calculus book for non-mathematicians. Given that you are getting theory from Spivak, there are probably sections you can glance through, but you should work every integration problem (most can be done in a couple of minutes). My openstax suggestion still stands or the first @micromass reference. If you live in a university town, you could get a cheap used copy of whatever they are using.
 
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  • #11
BTW, you do not need solutions for indefinite integrals. You can always take the derivative to check your work.
 
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1. What is the importance of integration in calculus?

Integration is a fundamental concept in calculus that allows us to find the area under a curve, calculate volumes of irregular shapes, and solve a variety of real-world problems. It is also necessary for understanding and applying other calculus concepts such as derivatives and differential equations.

2. What makes a calculus textbook's section on integration "good"?

A good section on integration should provide clear explanations of the fundamental concepts, a variety of examples with step-by-step solutions, and practice problems with varying levels of difficulty. It should also include applications of integration to real-world problems and connections to other calculus topics.

3. Are there any specific calculus textbooks that are known for their strong sections on integration?

Yes, there are several textbooks that are highly recommended for their coverage of integration, including "Calculus: Early Transcendentals" by James Stewart, "Thomas' Calculus" by George B. Thomas Jr. and Maurice D. Weir, and "Calculus" by Michael Spivak.

4. How can I improve my understanding of integration while using a textbook?

To improve your understanding of integration while using a textbook, it is important to actively engage with the material. This can include taking notes, working through practice problems, and seeking additional resources or explanations if needed. It may also be helpful to discuss concepts with classmates or seek assistance from a tutor.

5. Are there any online resources that can supplement a calculus textbook's section on integration?

Yes, there are many online resources that can supplement a textbook's section on integration. These can include video tutorials, interactive practice problems, and online calculators. Some popular websites for calculus help include Khan Academy, Wolfram Alpha, and Paul's Online Math Notes.

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