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.