A Note on Evaluating the Integral of Cos(1/4 arccos(x))

(10/28/2023)

Evaluate

where n is real number.

 

Solution

Let

Hence, we get

 

Substituting these into the given integral, we get

 

For n = 4, we obtain

 

Notes:

1. The Wolfram integral calculator provided a lengthy result for the integral of cos(1/4 arccos(x)) on August 28, 2023. See the results in Figure 1. Readers can also verify the results on the old versions of the Wolfram’s calculator early in 2024.

Figure 1

 

 
2. When readers/users encounter a particularly difficult problem (e.g., solving an equation, evaluating an integral), Wolfram’s online calculator may return a numeric result with a note stating,

Figure 2

 

This message often appears when the problem is unsolvable, giving the impression that Wolfram can solve it. To verify this, readers can input any unsolvable integral into the Wolfram calculator.
 
We have two points to discuss regarding this scenario:
 
a. Wolfram’s calculator may not have enough time to respond to a difficult problem, prompting the message, “Try again with Pro computation time.”
 
b. Wolfram’s calculator may not be able to provide a closed-form solution, returning only a numeric result. In such cases, it would be more transparent and respectful for Wolfram to inform readers/users, for example, “The solution cannot be solved at this time.” This approach would give credit to students or others who might later provide a solution. Eventually, Wolfram could learn the solution from these users or from published results. Mathematicians sometimes recognize a closed-form solution without detailed steps, and they can solve it. Wolfram should respect readers by acknowledging that not all problems are currently solvable, rather than implying they can solve any problem.
 
 
 
Note:
If readers try to check this integral, cos(1/4 arccos(x)), on Wolfram’s calculator today, it shows the "definite integral" with the correct result for a few seconds (in a flash) and then disappears. The solution for 'definite integral' of cos(1/4 arccos(x)) was previously unseen. This behavior appears similarly in other difficult integrals, which are understandably challenging at first. After giving an overall review, a friend jokes that it's a sneaky form of intellectual theft, even though it's not worth a dime at that moment. However, for Wolfram marketing, it's like accumulating a pocketbook of pennies, slowly but surely adding up.
(Updated on 10/8/2024)
 
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