MHB Definite integral of square root+cube root

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The integral of the square root and cube root posed in the discussion is challenging, and initial attempts at substitution were unsuccessful. WolframAlpha indicates the result is 6, but the user seeks a method to solve it using elementary functions. Participants emphasize the importance of showing progress in problem-solving to receive effective assistance. One suggested approach involves using a specific integral formula related to square roots. Ultimately, the integral may not be solvable by elementary functions alone.
tree21c
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Dear all,

Please solve this integral:

View attachment 7576

I tried integral by substitution, but failed.

Wolframalpha shows the result is 6, but I don't know how to proceed it.

Can it be solved by elementary function?
 

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Hello and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 
tree21c said:
Can it be solved by elementary function?

No; I don't think so.
 
Try applying the formula

$$\int\sqrt {x^2+a^2}dx=\frac{x\sqrt {x^2+a^2}}{2}+\frac{a^2}{2}ln(x+\sqrt {x^2+a^2})$$
 

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