How Do You Integrate 1/(cos(x)-1)?

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In summary, the conversation is about finding the integral of 1/(cosx-1), where the student first multiplied the function by cosx+1/cosx+1 and ended up with the integral of (cosx+1)/(-sinx)^2. They then discuss the integral of 1/(-sinx)^2 and determine it to be cotx. Finally, they mention the integral of cosx/(-sinx)^2 and believe it to be -(integral of cotx*cscx). The expert confirms that there is nothing wrong with the student's work and suggests checking any antiderivative by taking its derivative to ensure it is correct.
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montana111
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Homework Statement



help me out. I want to find the integral of 1/(cosx-1). I first multiplied the function by cosx+1/cosx+1. then i ended up with the integral of (cosx+1)/(-sinx)^2 leading to integral of cosx/(-sinx)^2 plus integral of 1/(-sinx)^2.

the integral of 1/(-sinx)^2 i believe is just cotx.

the integral of cosx/(-sinx)^2 i think is -(integral of cotx*cscx)


Homework Equations



integral of 1/(cosx-1)
 
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  • #2
What's your question? I don't see anything wrong in what you've described. You can check any antiderivative you get by taking its derivative. If you get back to the integrand, you're good.
 
  • #3
What function has as its derivative -cot x csc x?
 

Related to How Do You Integrate 1/(cos(x)-1)?

1. What is the formula for the integral of 1/(cosx-1)?

The formula for the integral of 1/(cosx-1) is:
∫(1/(cosx-1)) dx = ln|tan(x/2)| + C

2. What is the domain of the function 1/(cosx-1)?

The domain of the function 1/(cosx-1) is all real numbers except for x = 2πn, where n is an integer. This is because cosx-1 is undefined at these values, making the denominator 0.

3. Can the integral of 1/(cosx-1) be solved using substitution?

Yes, the integral of 1/(cosx-1) can be solved using the substitution u = tan(x/2). This will result in the integral becoming ∫(1/u) du, which can then be easily integrated to ln|u| + C.

4. Is there a specific method for solving the integral of 1/(cosx-1)?

Yes, the integral of 1/(cosx-1) can be solved using the method of partial fractions. This involves breaking down the function into simpler fractions and then integrating each part individually.

5. What is the significance of the integral of 1/(cosx-1) in mathematics?

The integral of 1/(cosx-1) has significance in various mathematical applications, such as in trigonometry and calculus. It can also be used to solve various differential equations and has applications in physics and engineering.

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