u = 1 + y⁴

I am going to assume that you mean ∫(0 to 1) 2/3pi*y^3√(1+y^4)dy

Use a u substitution by substituting 1+y^4 for u.

du/dy=4y^3 du=4y^3dy 1/4du=y^3dy

1/6pi ∫(1 to 2) √(u) du

Integrate: 1/6pi 2/3u^(3/2) from 1 to 2

1/6pi2/3(2)^(3/2)-1/6pi2/3(1)^(3/2)

your answer is (2pi√2-pi)/9 or .6382414692

http://www.wolframalpha.com/input/?i=0+to1+2%2F3+p...