How Can I Improve My Integration Techniques?

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    Integration
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Discussion Overview

The discussion focuses on improving integration techniques, particularly in the context of proper and improper Riemann integrals. Participants share their struggles with specific integrals and seek recommendations for resources and strategies to enhance their skills.

Discussion Character

  • Homework-related
  • Exploratory
  • Technical explanation

Main Points Raised

  • The original poster expresses difficulty with specific integrals and seeks guidance on improving integration skills and finding relevant books.
  • Some participants emphasize that practice is essential for mastering integration, suggesting that repeated exposure to various integrals will help develop skills and techniques.
  • One participant notes that certain integrals do not have indefinite integrals in terms of elementary functions, indicating a distinction between solvable and unsolvable integrals.
  • Another participant mentions a book with numerous exercises in Analysis 1 as a potential resource for practice.
  • There is a correction regarding the rendering of integrals in the original post, which some participants noted as problematic.

Areas of Agreement / Disagreement

Participants generally agree that practice is crucial for improving integration skills, but there is no consensus on specific methods or resources beyond general recommendations.

Contextual Notes

Some integrals discussed may not have solutions in terms of elementary functions, which could affect the approach to mastering integration techniques. The discussion also reflects varying levels of understanding and experience among participants.

Who May Find This Useful

Students and learners seeking to improve their integration skills, particularly in calculus and analysis, may find this discussion relevant.

pswongaa
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My integration skill is very poor, there a many integral that I can't solve, for example:
$$\int_{0}^{\infty}\frac{\sin^{2n+1} x}{x} dx$$
$$\int_{0}^{\infty}\frac{\cos ax-\cos bx}{x} dx$$
but my friend could solve them very quickly, so may I wonder if there are any books about technique for proper and improper riemann integral? Also what is the best way to master integration?
 
Last edited by a moderator:
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pswongaa said:
My integration skill is very poor, there a many integral that I can't solve, for example:
$$\int_{0}^{\infty}\frac{\sin^{2n+1} x}{x} dx $$
$$\int_{0}^{\infty}\frac{\cos ax-\cos bx}{x} dx $$
but my friend could solve them very quickly, so may I wonder if there are any books about technique for proper and improper riemann integral? Also what is the best way to master integration?

Something is wrong with your post, the integrals aren't rendering correctly.

Mod note: They're fixed now. They were missing the LaTeX tags and were also malformed. I made corrections, but I'm not 100% certain of what the OP intended.
 
Last edited by a moderator:
jedishrfu said:
Something is wrong with your post, the integrals aren't rendering correctly.

the integrals themselves are not important, I just want to find a way to improve my integration skill
 
Unfortunately the only way to get good (ie. fast and accurate) at integrals is by doing integrals - a lot of them.

By going through the procedure repeatedly you will build a collection of tricks that you will learn how to use and when to apply them.

You should open up any calc book and start doing every integral. When you get stuck ask others or look for solutions or
Hints.
 
Some integrals can be solved (they have indefinite integrals which are composed of elementary functions) and there are many more which cannot be solved, although the definite integrals can be shown to be equal to a certain value.

It just so happens that the two examples you chose do not have indefinite integrals which are composed of elementary functions.
 
pswongaa said:
Also what is the best way to master integration?

Integrate! As much as possible.

There's this book that has thousands of exercises in Analysis 1.
 

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