Evaluate the definite integral from x=0 to x=1
the cubed root of (1 + 7x) dx

Use a  substitution of u = 7x+1 then du = 7 dx and dx = 1/7 du 
so f(x) = cube root (u) 1/7 du 
integral 1/7 (u)^1/3 du 
using simple power rules gives us 
1/7 (3/4) u^4/3 
which equals 3/28 u^4/3 
replacing u with 7x+1 gives us 3/28 (7x+1)^4/3 
just plug in 0 and 1 and subtract. 
3/28(8)^4/3 - 3/28 (1)^4/3 
which is 3/28(16) - 3/28 = 3/28 * 15 
so final answer is 45/28.