Integrating cot^4 x (csc^4 x) dx Using Identities and U Substitution

In summary, using identities and u substitution is beneficial when integrating cot^4 x (csc^4 x) dx as it simplifies the expression and makes integration easier. The Pythagorean identity can be used to rewrite the expression and u substitution can further simplify and integrate it. Other trigonometric identities such as the double angle and half angle identities can also be used. An example of a problem where integrating this expression would be useful is in calculating the work done by a force on a particle moving along a circular path.
  • #1
johnhuntsman
76
0
∫(cot^4 x) (csc^4 x) dx

Wolfram wants to use the reduction formula, but I'm meant to do this just using identities and u substitution. I was thinking something along the lines of:

=∫cot^4 x (cot^2 x + 1)^2 dx

=∫cot^8 x + 2cot^6 x + cot^4 x dx

but I don't know where to go from there.
 
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  • #2
Try ∫(cot^4 x) (1+cot^2 x) (csc^2 x) dx

Whats the derivative of cot(x)?
 
  • #3
Thanks. Worked it out with that in mind. I appreciate it : D
 

Related to Integrating cot^4 x (csc^4 x) dx Using Identities and U Substitution

1. What is the purpose of using identities and u substitution when integrating cot^4 x (csc^4 x) dx?

The purpose of using identities and u substitution is to simplify the expression and make it easier to integrate. By using trigonometric identities, the expression can be rewritten in a more manageable form. U substitution allows for the integration of more complex functions by substituting a variable that allows for easier integration.

2. How do you use the Pythagorean identity when integrating cot^4 x (csc^4 x) dx?

The Pythagorean identity, csc^2 x = 1 + cot^2 x, can be used to rewrite the expression as cot^4 x (1 + cot^2 x)^2 dx. Then, using u substitution, let u = cot x, the expression can be further simplified and integrated.

3. Can you explain the steps for integrating cot^4 x (csc^4 x) dx using identities and u substitution?

Step 1: Rewrite the expression using the Pythagorean identity csc^2 x = 1 + cot^2 x.
Step 2: Expand the expression to get cot^4 x (1 + 2cot^2 x + cot^4 x) dx.
Step 3: Let u = cot x, then du = -csc^2 x dx.
Step 4: Substitute u and du into the expression to get the integral of -u^2 (1 + 2u^2 + u^4) du.
Step 5: Integrate the expression using the power rule for integration.
Step 6: Substitute u back in for cot x to get the final answer.

4. Are there any other trigonometric identities that can be used to integrate cot^4 x (csc^4 x) dx?

Yes, there are multiple trigonometric identities that can be used, such as the double angle identity, csc 2x = 2cot x csc x, and the half angle identities, csc^2 x = 1/2 (1 + cos 2x) and cot^2 x = 1/2 (1 - cos 2x).

5. Can you give an example of a problem where integrating cot^4 x (csc^4 x) dx would be useful?

Integrating cot^4 x (csc^4 x) dx can be useful in solving problems involving the area under the curve of a function that contains cotangent and cosecant terms. For example, in calculating the work done by a force on a particle moving along a circular path, the work is equal to the line integral of the force vector along the path. If the force vector is expressed in terms of cotangent and cosecant, integrating cot^4 x (csc^4 x) dx can help find the work done.

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