What is the indefinite integral of the function y=(4x^3+1)square root(x^4+x)?

We'll use substitution to evaluate the indefinite
integral.

Let x^4 + x =
t

We'll differentiate both sides and we'll
get:

(4x^3 + 1)dx = dt

Int
(4x^3 + 1)*sqrt(x^4 + x)dx = Int (sqrt t)dt

Int (sqrt t)dt
= Int t^(1/2) dt

Int t^(1/2) dt = t^(1/2 + 1)/(1/2 + 1) +
C

Int t^(1/2) dt = 2t^(3/2)/3 +
C

Int t^(1/2) dt = [2t*sqrt(t)]/3 +
C

Therefore, the indefinite integral of the
given function is Int (4x^3 + 1)*sqrt(x^4 + x)dx = [2*(x^4 + x)*sqrt(x^4 + x)]/3 +
C.