dualogia

g(x)+h(y)=f(x)


x+y = 0 <==> y=(-x)
x+y=(x+(-a))^{n} <==> ( x=a <==> y=(-a) )
x+y=(x^{p}+(-a)) <==> ( x=a^{(1/p)} <==> y=(-1)a^{(1/p)} )
x+y=(e^{px}+(-a)) <==> ( x=ln(a^{(1/p)}) <==> y=(-1)ln(a^{(1/p)}) )
x+y=(ln(px)+(-a)) <==> ( x=e^{a+(-1)ln(p)} <==> y=(-1)e^{a+(-1)ln(p)}) )
x+y=( (a^{(k/p)}x^{p+(-k)})+(-a) ) <==> ( x=a^{(1/p)} <==> y=(-1)a^{(1/p)} )
x+y=(e^{p·x^{m}}+(-a)) <==> ( x=( ln(a^{(1/p)}) )^{(1/m)} <==> y=(-1)( ln(a^{(1/p)}) )^{(1/m)} )


x^{2}+y^{2} = 0 <==> y=ix or y=(-i)x
x^{n}+y^{n} = 0 <==> y=e^{(1/n)·pi·i}x or y=e^{(-1)(1/n)·pi·i}x


x^{n+1}+y^{n+1} = n^{n+1} <==> ( x=cos[n](t) <==> y=sin[n](t) )
x^{n+1}+y^{n+1} = 1 <==> ( x=(cos[n](t)/n) <==> y=(sin[n](t)/n) )






d_{x}[g(x)]+d_{y}[h(y)]d_{x}[y]=d_{x}[f(x)]


d_{y}[h(y)]d_{x}[y]=d_{x}[f(x)]+(-1)d_{x}[g(x)]
d_{y}[h(y)]d_{x}[y]=d_{x}[f(x)+(-1)g(x)]
d_{y}[h(y)]d_{x}[y]=d_{x}[h(y)]


int[g(x)]d[x]·d_{x}[y]+int[h(y)]d[y]=int[f(x)]d[x]·d_{x}[y]


int[h(y)]d[y]=( int[f(x)]d[x]+(-1)int[g(x)]d[x] )·d_{x}[y]
int[h(y)]d[y]=int[f(x)+(-1)g(x)]d[x]·d_{x}[y]
int[h(y)]d[y]=int[h(y)]d[x]·d_{x}[y]
int[h(y)]d[y]=int[h(y)]d[y]


F(x)=g(x)h(y)
F(x)=g(x)f(x)+(-1)( g(x) )^{2}
f(x)=( F(x)/g(x) )+g(x)
g(x)+h(y)=( F(x)/g(x) )+g(x)
h(y)=( F(x)/g(x) )
g(x)h(y)=F(x)

d_{x}[F(x)]=d_{x}[g(x)]f(x)+g(x)d_{x}[f(x)]+(-2)g(x)d_{x}[g(x)]
d_{x}[g(x)h(y)]=d_{x}[g(x)]f(x)+g(x)d_{x}[f(x)]+(-2)g(x)d_{x}[g(x)]
d_{x}[g(x)]h(y)+g(x)d_{x}[h(y)]=d_{x}[g(x)]( f(x)+(-1)g(x) )+g(x)d_{x}[f(x)+(-1)g(x)]
d_{x}[g(x)]h(y)+g(x)d_{x}[h(y)]=d_{x}[g(x)]h(y)+g(x)d_{x}[h(y)]


int[F(x)]d[x] = int[g(x)]d[x] [o(x)o] int[h(y)]d[x]
int[F(x)]d[x] = int[g(x)]d[x] [o(x)o] int[f(x)]d[x]+(-1)int[g(x)]d[x] [o(x)o] int[g(x)]d[x]
int[F(x)]d[x] = int[g(x)]d[x] [o(x)o] ( int[f(x)]d[x]+(-1)int[g(x)]d[x] )
int[F(x)]d[x] = int[g(x)]d[x] [o(x)o] ( int[f(x)+(-1)g(x)]d[x] )
int[F(x)]d[x] = int[g(x)]d[x] [o(x)o] int[h(y)]d[x]




x+y=0
int[y]d[y]=int[(-x)]d[(-x)]
int[y]d[y]=(1/2)(-x)^{2}
d_{x}[y]=d_{x}[(-x)]
d_{x}[y]=(-1)
F(x)=(-1)x^{2}


x+y=(x+(-a))^{n}
int[y]d[y]=int[(x+(-a))^{n}+(-x)]d[(x+(-a))^{n}+(-x)]
int[y]d[y]=(1/2)( (x+(-a))^{n}+(-x) )^{2}
d_{x}[y]=d_{x}[(x+(-a))^{n}+(-x)]
d_{x}[y]=n(x+(-a))^{(n+(-1))}+(-1)
F(x)=x(x+(-a))^{n}+(-1)x^{2}
F(a)=(-1)a^{2}


x+y=(x^{p}+(-a))
int[y]d[y]=int[(x^{p}+(-a))+(-x)]d[(x^{p}+(-a))+(-x)]
int[y]d[y]=(1/2)( (x^{p}+(-a))+(-x) )^{2}
d_{x}[y]=d_{x}[(x^{p}+(-a))+(-x)]
d_{x}[y]=px^{(p+(-1))}+(-1)
F(x)=(x^{p+1}+(-a)x)+(-1)x^{2}
F(a^{(1/p)})=(-1)( a^{(1/p)} )^{2}