We have dy/dx = 4x^3 + 4x.
dy/dx = 4x^3 + 4x
=> dy = (4x^3 + 4x) dx
Integrate both the sides
Int [ dy ] = Int [ (4x^3 + 4x) dx ]
=> y = 4x^4 / 4 + 4*x^2 / 2
=> y = x^4 + 2*x^2 + C
As the curve passes through (1 , 4)
4 = 1^4 + 2*1^2 + C
=> 4 = 1 + 2 + C
=> C = 1
This gives y = x^4 + 2*x^2 + 1
The required equation of the curve is y = x^4 + 2*x^2 + 1