indefinite integrals

[Someonewithoutanswers]

I'm stuck, I don't know how to solve this integral ... please help me!
Between the brackets there is the solution.

 

[Dr.Peterson]

Please tell us what methods you have learned, and show your attempt at one method, so we can see where you are stuck. That's how we work.

I haven't tried it yet, but the presence of both square root and cube root suggest making the substitution [MATH]u = \sqrt[6]{x}[/MATH], so that both are powers of u. The answer supports that idea.

 

[Someonewithoutanswers]

Please tell us what methods you have learned, and show your attempt at one method, so we can see where you are stuck. That's how we work.

I haven't tried it yet, but the presence of both square root and cube root suggest making the substitution [MATH]u = \sqrt[6]{x}[/MATH], so that both are powers of u. The answer supports that idea.

I had already tried to integrate by parts ( Sf'(x)g(x) = f(x)g(x) - Sf(x)g'(x) ) and to integrate by substitution but it led me nowhere. Now, thanks to your suggestion I made some progresses and I think I should decompose the function to be able to use the following formula:
but I'm having trouble decomposing the function.

 

[Dr.Peterson]

This isn't what I call "showing your work", only describing it in general terms. I have no idea whether you have made any mistakes we could help you fix.