Complicated double integral: int[0,1][3(x-1)^2 {int[0,x][sqrt{1-(t-1)^4} dt) dx

[amainiero]

Friend of mine asked me to help him on a challenge problem he was given by his professor.

. . . . .\(\displaystyle \displaystyle \int_0^1\, 3\, (x\, -\, 1)^2\, \left(\int_0^x\, \sqrt{1\, -\, (t\, -\, 1)^4\,}\, dt\right)\, dx\)

I'm not even sure where to start on this- I tried focusing on the inner part, which became a sqrt(1-u^4), but every attempt I made at breaking that into something useful failed me. I tried going for trig subs, u-subs, parts, and a very failed PFD which all led to nothing. Wolfram was unusually unhelpful- yielding some crazy result in terms of undefined functions, and all the other go-to's I tried also didn't work. Hoping somebody here can figure it out.

I figure there is some sneaky trick a more keen-eyed person will see on this.

 

[stapel]

Friend of mine asked me to help him on a challenge problem he was given by his professor.

. . . . .\(\displaystyle \displaystyle \int_0^1\, 3\, (x\, -\, 1)^2\, \left(\int_0^x\, \sqrt{1\, -\, (t\, -\, 1)^4\,}\, dt\right)\, dx\)

I'm not even sure where to start on this- I tried focusing on the inner part, which became a sqrt(1-u^4), but every attempt I made at breaking that into something useful failed me. I tried going for trig subs, u-subs, parts, and a very failed PFD which all led to nothing. Wolfram was unusually unhelpful- yielding some crazy result in terms of undefined functions, and all the other go-to's I tried also didn't work. Hoping somebody here can figure it out.

I figure there is some sneaky trick a more keen-eyed person will see on this.

Unfortunately, we have learned (the hard way) that attempting to communicate through a translator who doesn't speak the language is fated to failure. So please have your friend provide a clear listing of his thoughts and efforts so far, along with recent topics of study (as this will help us figure out what method is probably expected for this solution). Thank you!