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.
