Solve Impossible Integral: Guide to \int dx [0,1]

[Sebobas]

Homework Statement

I have to find the definite integral of: $\int\left(\sqrt[3]{1-x^{7}}-\sqrt[7]{1-x^{3}}\right)dx$ with bounds [0,1]

2. The attempt at a solution

I know that this can be done with hypergeometric functions, but I ca't use them because "we haven't seem them yet", so I have to do this with "everyday" integration tools. The only thing I was told was that I have to start with a substitution and the last part would be a cyclical integral, which tells me that I probably have to also use integration by parts in the middle.

I'm not asking for someone to give me the answer right away, I just need a starting point/mini-guide to help me get to the answer on my own, although I'm grateful with any kind of help (: Thanks!

 

[clamtrox]

Which substitutions have you tried so far?

 

[Sebobas]

Which substitutions have you tried so far?

u= x⁷, x³, 1-x³, 1-x⁷

 

[Curious3141]

Homework Statement

I have to find the integral of: $\int\left(\sqrt[3]{1-x^{7}}-\sqrt[7]{1-x^{3}}\right)dx$

2. The attempt at a solution

I know that this can be done with hypergeometric functions, but I ca't use them because "we haven't seem them yet", so I have to do this with "everyday" integration tools. The only thing I was told was that I have to start with a substitution and the last part would be a cyclical integral, which tells me that I probably have to also use integration by parts in the middle.

I'm not asking for someone to give me the answer right away, I just need a starting point/mini-guide to help me get to the answer on my own, although I'm grateful with any kind of help (: Thanks!

Sure that's not a definite integral with bounds of [0,1]?

Because there's a neat trick to immediately and trivially evaluate it in that case. See my earlier post in this thread: https://www.physicsforums.com/showthread.php?t=571323

 

[Infinitum]

Sure that's not a definite integral with bounds of [0,1]?

Because there's a neat trick to immediately and trivially evaluate it in that case. See my earlier post in this thread: https://www.physicsforums.com/showthread.php?t=571323

Wow, that's a really nice trick.

 

[Sebobas]

Sure that's not a definite integral with bounds of [0,1]?

Because there's a neat trick to immediately and trivially evaluate it in that case. See my earlier post in this thread: https://www.physicsforums.com/showthread.php?t=571323

It is! Sorry, I thought I put the bounds on the integral. I'm new here so I probably didn't do it right...fixed the post (: and thanks for that trick!