Integrate (cos^2(x))/(sin^6(x))

  • Thread starter n77ler
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In summary, the problem is to integrate (cos^2(x))/(sin^6(x)) and there are two possible methods suggested - using double angles or u-substitution. After some discussion, the solution is found to be cosec^6(x) - cosec^4(x). However, the person asking for help ends up using a different, simpler method of separating the integral into two parts and solving.
  • #1
n77ler
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[SOLVED] ff

Homework Statement



integrate (cos^2(x))/(sin^6(x))

Homework Equations



Double angles?
u-sub?

The Attempt at a Solution



I tried using double angles but I am really not sure what to do I havnt seen any like this... I have only done stuff without fraction like integrating cos^3x sin^5x can anyone just start me off please
 
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  • #2
Hi n77ler! :smile:

Hint: it's cosec^6(x) - cosec^4(x).

Does that help … ? :smile:
 
  • #3
hmm kind of I am not sure where you got that though. The division in an integral can be changed to subtration i guess... but I am not sure how you are getting the two csc's. I know it is 1/sin but not sure other than that
 
  • #4
cos^2(x)/sin^6(x)

= (1 - sin^2(x))/sin^6(x) = 1/sin^6(x) - 1/sin^4(x)

= cosec^6(x) - cosec^4(x). :smile:
 
  • #5
right, god lol I'm not functioning well right now.
 
  • #6
ok so i separate those into two different integrals and then integrate right
 
  • #7
yup :)
 
  • #8
meh gave up cause it was to much parts so i went back to start and found an easier way lol...

(cos^2x/sin^2x ) (1/sin^4x) and then cot^2xcsc^4x and solved...thanks tho :)
 

1. What is the integration of (cos^2(x))/(sin^6(x))?

The integration of (cos^2(x))/(sin^6(x)) is equal to -1/5cot^5(x) + C.

2. How do you solve the integration of (cos^2(x))/(sin^6(x))?

To solve the integration of (cos^2(x))/(sin^6(x)), you can use the trigonometric identity cos^2(x) = 1/2(1+cos(2x)) and substitute it into the integral. Then, use trigonometric substitution to simplify the integral and solve for the final answer.

3. Can the integration of (cos^2(x))/(sin^6(x)) be simplified further?

Yes, the integration of (cos^2(x))/(sin^6(x)) can be simplified further by using trigonometric identities and algebraic manipulation. The final answer can be written in different forms, such as using trigonometric functions or inverse trigonometric functions.

4. What is the domain of the integral of (cos^2(x))/(sin^6(x))?

The domain of the integral of (cos^2(x))/(sin^6(x)) is all real numbers except for values of x where sin(x) = 0. This is because dividing by 0 is undefined, and the integral cannot be evaluated at those points.

5. Is there a shortcut to solve the integration of (cos^2(x))/(sin^6(x))?

There is no specific shortcut to solve the integration of (cos^2(x))/(sin^6(x)). However, knowing the relevant trigonometric identities and being familiar with trigonometric substitution methods can make the integration process faster and more efficient.

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