Find the antiderivative of y= square root(1+square rootx).

To find the antiderivative of the given function, we'll
have to determine the indefinite integral.

Int  sqrt(1+sqrt
x) dx

We'll substitute 1 + sqrt x =
t

We'll differentiate both
sides:

dx/2sqrt x = dt

dx = 2
sqrt x*dt

But sqrt x = t -
1

dx = 2(t - 1)dt

We'll
re-write the integral in t:

Int sqrt t*2(t - 1)dt = 2Int
sqrt t^3 dt - 2 Int sqrt t dt

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

 Int
sqrt t*2(t - 1)dt = 4t^(5/2)/5 - 4t^(3/2)/3 + C

 Int sqrt
t*2(t - 1)dt = 4t^(3/2)( t/5 - 1/3) +
C

Int sqrt(1+sqrt x)dx = 4(1 + sqrt
x)^(3/2)[(1 + sqrt x)/5 - 1/3] + C