Am I doing this integration problem right?

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The integration problem involves the expression e^(x^(5/3)), which is complex and challenging to solve. The integral of x^(5/3) is known to be (3/8)x^(8/3), but the presence of e complicates the integration process. Clarification is needed on whether the expression is (e^x)^(5/3) or e^(x^(5/3)), as the former is simpler to handle. The latter is significantly more difficult and may not be solvable using basic calculus techniques. Overall, the discussion emphasizes the need for careful interpretation of the expression to determine the appropriate integration approach.
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Homework Statement


Hello, I am struggling with an integration problem.

e ^ x ^ (5/3)

Homework Equations



Um.

The Attempt at a Solution



So I know the integral of x^(5/3) is (3/8)x^(8/3). But when I am trying to integrate the whole thing, with e in it, do I just treat x^(5/3) like a constant? Or do I still have to integrate it?

Possible solutions: e^x^(5/3)
or
e^(3/8)x^(8/3)
or
something else... please help...
 
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Take a close look at the problem. Does it appear to be (e^x)^(5/3) or is it e^(x^(5/3))?

The first is relatively easy (simplify and substitute). If in doubt, I'd assume this interpretation.

The second is really, really ugly. I can't figure out how to do it, my TI-89 chokes on it, and Maple spits out the following gibberish:
output.png

so I'm guessing it's not really solvable by hand using first/second-year calculus techniques.
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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