What is the solution for (dy)/(dx)=4x+(9x^2)/((3x^3+1)^{3/2)} ?

The solution for (dy)/(dx)=4x+(9x^2)/((3x^3+1)^{3/2)} is y=2x^2-2/((3x^3+1)^{1/2)}+c_{1}

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(dy)/(dx)=4x+(9x^2)/((3x^3+1)^{3/2)}

\frac{d y}{d x} = 4 x + \frac{9 x ^{2}}{( 3 x ^{3} + 1 ) ^{\frac{3}{2}}}

y = 2 x^{2} - \frac{2}{( 3 x ^{3} + 1 ) ^{\frac{1}{2}}} + c_{1}

Solution steps

\frac{d y}{d x} = 4 x + \frac{9 x ^{2}}{( 3 x ^{3} + 1 ) ^{\frac{3}{2}}}

Substitute

\frac{d y}{d x}

with

y^{'} (x)

y^{^{'}} (x) = 4 x + \frac{9 x ^{2}}{( 3 x ^{3} + 1 ) ^{\frac{3}{2}}}

f^{^{'}} (x) = g (x)

then

f (x) = \int g (x) d x

y = \int 4 x + \frac{9 x ^{2}}{( 3 x ^{3} + 1 ) ^{\frac{3}{2}}} d x

\int 4 x + \frac{9 x ^{2}}{( 3 x ^{3} + 1 ) ^{\frac{3}{2}}} d x = 2 x^{2} - \frac{2}{( 3 x ^{3} + 1 ) ^{\frac{1}{2}}} + c_{1}

y = 2 x^{2} - \frac{2}{( 3 x ^{3} + 1 ) ^{\frac{1}{2}}} + c_{1}

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What is the solution for (dy)/(dx)=4x+(9x^2)/((3x^3+1)^{3/2)} ?
The solution for (dy)/(dx)=4x+(9x^2)/((3x^3+1)^{3/2)} is y=2x^2-2/((3x^3+1)^{1/2)}+c_{1}