What is the solution for (dy)/(dx)=sin(2x)-1/2 sin(4x) ?

The solution for (dy)/(dx)=sin(2x)-1/2 sin(4x) is y=-1/2 cos(2x)+1/8 cos(4x)+c_{1}

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(dy)/(dx)=sin(2x)-1/2 sin(4x)

\frac{d y}{d x} = sin (2 x) - \frac{1}{2} sin (4 x)

y = - \frac{1}{2} cos (2 x) + \frac{1}{8} cos (4 x) + c_{1}

Solution steps

\frac{d y}{d x} = sin (2 x) - \frac{1}{2} sin (4 x)

Substitute

\frac{d y}{d x}

with

y^{'} (x)

y^{^{'}} (x) = sin (2 x) - \frac{1}{2} sin (4 x)

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

then

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

y = \int sin (2 x) - \frac{1}{2} sin (4 x) d x

\int sin (2 x) - \frac{1}{2} sin (4 x) d x = - \frac{1}{2} cos (2 x) + \frac{1}{8} cos (4 x) + c_{1}

y = - \frac{1}{2} cos (2 x) + \frac{1}{8} cos (4 x) + c_{1}

f (x) = x (1 - 2 x)^{3}, (1, - 1)

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What is the solution for (dy)/(dx)=sin(2x)-1/2 sin(4x) ?
The solution for (dy)/(dx)=sin(2x)-1/2 sin(4x) is y=-1/2 cos(2x)+1/8 cos(4x)+c_{1}