Integrating 1/(1+a*cosx): A Step-by-Step Guide

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In summary, "Integrating 1/(1+a*cosx): A Step-by-Step Guide" means breaking down the process of finding the integral of the function 1/(1+a*cosx) into smaller, manageable steps. This topic is important for its applications in mathematics and physics and for developing problem-solving skills. Prerequisites for understanding this guide include knowledge of algebra, trigonometry, and calculus, as well as integration techniques. A helpful tip for approaching this type of integration problem is to rewrite the function in terms of a single trigonometric function and to carefully track substitutions and changes in limits of integration. Common mistakes to avoid include forgetting to use the chain rule and checking for algebraic mistakes and changes in limits of integration
  • #1
tennishaha
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Homework Statement



How to integral this one?

1/(1+a*cosx)



Homework Equations





The Attempt at a Solution

 
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  • #2
In this case, you use wolframalpha.com because the calculation by hand is brutal.

http://www.wolframalpha.com/input/?i=integrate+(1%2Ba*cos(x))^-1

Some people think that's cheating, but I think doing the grunt work here would be a waste of time since there's no new insight. From the looks of it, it's just several cumbersome integration by parts with the inverse hyperbolic tangent function.
 
  • #3
I would try [tex]t=\tan (\theta /2)[/tex], it's one of the standard ones.
 

1. What does "Integrating 1/(1+a*cosx): A Step-by-Step Guide" mean?

"Integrating" refers to the process of finding the integral of a mathematical function. In this case, we are looking specifically at the function 1/(1+a*cosx). The "Step-by-Step Guide" indicates that we will be breaking down the integration process into smaller, manageable steps.

2. Why is this topic important?

Integrating 1/(1+a*cosx) is a common problem in mathematics and physics, so understanding how to solve it can be useful in various applications. It also helps to develop problem-solving skills and improve understanding of mathematical concepts.

3. What are the prerequisites for understanding this guide?

A basic understanding of algebra, trigonometry, and calculus is necessary to follow this guide. Familiarity with integration techniques such as substitution and integration by parts will also be beneficial.

4. Are there any tips for approaching this type of integration problem?

Yes, a helpful tip is to rewrite the function in terms of a single trigonometric function. In this case, we can use the identity cosx = sin^2(x/2) to simplify the integral. It is also important to carefully track substitutions and any resulting changes in limits of integration.

5. Are there any common mistakes to avoid?

One common mistake when integrating functions like 1/(1+a*cosx) is to forget to use the chain rule when differentiating the inside function. It is also important to check for algebraic mistakes and to carefully evaluate the limits of integration after substitutions have been made.

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