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

  • Thread starter Thread starter n77ler
  • Start date Start date
  • Tags Tags
    Integrate
Click For Summary
SUMMARY

The integral of (cos^2(x))/(sin^6(x)) can be simplified to cosec^6(x) - cosec^4(x) through algebraic manipulation. The transformation involves rewriting the expression as (1 - sin^2(x))/sin^6(x), which separates into two integrals. The discussion highlights the use of trigonometric identities and substitution methods, specifically focusing on the relationship between sine and cosecant functions. The final approach suggested involves expressing the integral in terms of cotangent and cosecant functions for easier integration.

PREREQUISITES
  • Understanding of trigonometric identities, specifically sine and cosecant functions.
  • Familiarity with integration techniques, including substitution and separation of integrals.
  • Knowledge of algebraic manipulation of fractions in calculus.
  • Experience with double angle formulas in trigonometry.
NEXT STEPS
  • Study the integration of trigonometric functions using substitution methods.
  • Learn about the properties and applications of cosecant and cotangent functions in calculus.
  • Explore advanced techniques for integrating rational functions involving trigonometric identities.
  • Practice problems involving the integration of products of sine and cosine functions.
USEFUL FOR

Students studying calculus, particularly those focusing on integration techniques involving trigonometric functions, as well as educators looking for examples of integrating complex trigonometric expressions.

n77ler
Messages
89
Reaction score
0
[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
 
Last edited:
Physics news on Phys.org
Hi n77ler! :smile:

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

Does that help … ? :smile:
 
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
 
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:
 
right, god lol I'm not functioning well right now.
 
ok so i separate those into two different integrals and then integrate right
 
yup :)
 
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 :)
 

Similar threads

Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x
  • · Replies 3 ·
Replies
3
Views
2K
Calculating radius of gyration of plane figure about x-axis
  • · Replies 5 ·
Replies
5
Views
2K
Find the result of definite integration
  • · Replies 6 ·
Replies
6
Views
2K
Differentiate the given integral
  • · Replies 15 ·
Replies
15
Views
2K
Solve the problem involving the given double integral
  • · Replies 1 ·
Replies
1
Views
2K
Integrate (xsinx - cosx)/x^2 with Intro Calc Techniques
  • · Replies 3 ·
Replies
3
Views
2K
##\lim_{x \to0} \left(\dfrac{1}{\sin^2 x}-\dfrac{1}{x^2}\right)##
  • · Replies 5 ·
Replies
5
Views
2K
Why does dividing by ##\sin^2 x## solve the integral?
  • · Replies 27 ·
Replies
27
Views
4K
Differentiate ##f(x)=x\cos{x}+2\tan{x}: D/dx ##\tiny{2.4.2}##
  • · Replies 10 ·
Replies
10
Views
2K
The integral of (sin x + arctan x)/x^2 diverges over (0,∞)
  • · Replies 5 ·
Replies
5
Views
2K