Integrate x^3/(x^5-1): Solutions

In summary, the conversation was about a question and answer regarding an integral. The answer was provided in a complicated form and the person asking for clarification wanted to know if it could be solved without a calculator. The expert advised leaving it to a computational software unless the person has a special interest in tackling difficult integrals.
  • #1
Andy_ToK
43
0
Hi, here is the question
integrate x^3/(x^5-1)
 
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  • #2
Hi, here is the answer:

(2 Sqrt[2 (5 + Sqrt[5])]

1 - Sqrt[5] + 4 x
ArcTan[---------------------] +
Sqrt[2 (5 + Sqrt[5])]

2 Sqrt[10 - 2 Sqrt[5]]

1 + Sqrt[5] + 4 x
ArcTan[--------------------] +
Sqrt[10 - 2 Sqrt[5]]

4 Log[-1 + x] +

(-1 + Sqrt[5])

(-1 + Sqrt[5]) x 2
Log[1 - ---------------- + x ] -
2

(1 + Sqrt[5])

(1 + Sqrt[5]) x 2
Log[1 + --------------- + x ]) / 20
2Edit: The format didn't come out too well. Go to http://integrals.wolfram.com/index.jsp and do it yourself.
 
  • #3
hi, thanks. I have the answer but wonder how to solve it without using calculator. Sorry if I wasn't clear.
 
  • #4
Does it look like it would be easy?

My advice- leave it to Mathematica unless you have a pathological interest in doing near-impossible integrals (like GibZ).
 

1. What is the general approach to solving integrals?

The general approach to solving integrals is to first identify the appropriate integration technique, such as u-substitution or integration by parts. Then, use the corresponding formula or method to integrate the function. Finally, add any necessary constants of integration.

2. How do I apply u-substitution to this integral?

To apply u-substitution, first choose a variable u and then find its derivative du. Then, substitute u and du into the integral, making sure to also adjust the limits of integration if necessary. Finally, solve the resulting integral in terms of u and then substitute back in for x.

3. Can I use integration by parts for this integral?

Yes, integration by parts can be used for this integral. Choose the appropriate u and dv based on the formula 𝑢𝑑𝑣 = 𝑢𝑑𝑣 − ∫𝑑𝑢𝑣. Then, integrate dv and differentiate u, and substitute them into the formula to solve for the integral.

4. How do I handle improper integrals?

To handle improper integrals, first check if the integral is convergent or divergent. If it is convergent, use the limit definition of the integral to solve it. If it is divergent, use the comparison test or the limit comparison test to determine the behavior of the integral.

5. What is the process for solving integrals with trigonometric functions?

The process for solving integrals with trigonometric functions involves using trigonometric identities and substitution to simplify the integral. Then, apply the appropriate integration techniques, such as u-substitution or integration by parts, to solve the integral. Finally, use the inverse trigonometric functions to evaluate the integral in terms of x.

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