plz help thank you end SolutionThis is of the form MdxNdy0 M

plz help thank you

end

This is of the form

Mdx+Ndy=0

M=4x^3-y^2 sin(x)

N=2y cos(x)+12y^2

M_y=-2y sin(x)

N_x=-2y sin(x)=M_y

SUbscript denotes partial derivative

HEnce we ahve an exact differential

So the integral is of teh form

F(x,y)=C, C is arbitrary constant

So that

dF=Mdx+Ndy

M=F_x,N=F_y

F_x=4x^3-y^2 sin(x)

INtegrating w.r.t. x treating y as a constant gives

F=x^4+y^2 cos(x)+g(y)

So,

F_y=2y cos(x)+g\'(y)=N=2y cos(x)+12y^2

So, g\'(y)=12y^2

g(y)=4y^3

SO,

F=x^4+y^2 cos(x)+4y^3