Find the equation of a curve that passes through the point (1;4) and dy/dx=4x^3+4x

To determine the equation of the curve, we'll have to
determine the antiderivative of the given expression.

Int
dy = Int (4x^3+4x)dx

We'll use the property of integrals to
be additive:

Int (4x^3+4x)dx = Int 4x^3 dx + Int
4xdx

Int (4x^3+4x)dx = 4 Int x^3dx + 4Int
xdx

Int (4x^3+4x)dx = 4*x^4/4 + 4*x^2/2 +
C

We'll simplify and we'll
get:

Int (4x^3+4x)dx = x^4 + 2x^2 +
C

The expression represents a family of curves that depends
on the values of the constant C.

We know, from enunciation
that the point  (1 , 4) is located on the curve. Therefore it's coordinates will verify
the equation of the curve.

4 = (1)^4 +2*(1)^2 +
C

4 = 1 + 2 + C

4 = 3 +
C

C = 4 - 3

C =
1

The equation of the curve, whose derivative
is dy/dx=4x^3+4x , is the complete square: y = x^4 + 2x^2 + 1 = (x^2 +
1)^2.