Given f'(x)=(3x+1)^(-1/5), what is f(x)?

We'll take the inverse operation and we'll integrate the given
function:

Int (3x+1)^(-1/5) dx =
f(x)

We'll replace the expression 3x + 1 by the variable
t.

3x + 1 = t

We'll differentiate both
sides:

3dx = dt

dx =
dt/3

We'll re-write the integral having as variable
t:

Int (3x+1)^(-1/5) dx = Int
t^(-1/5)*(dt/3)

We'll evaluate the
integral

Int t^(-1/5)*(dt/3) = (1/3)*t^(-1/5 + 1)/(-1/5 + 1) +
C

Int t^(-1/5)*(dt/3) = (1/3)*t^(4/5)/(4/5) +
C

Int t^(-1/5)*(dt/3) = 5*t^(4/5)/12 +
C

The requested function is f(x) = 5*(3x+1)^(4/5)/12 +
C