calculo integral

int[ ( g(f(x)) )^{n} ] d[x] = ...

... (1/(n+1))·( g(f(x)) )^{n+1} [o(x)o] ( g(f(x)) )^{[o(x)o](-1)}

int[ ( f(x) )^{n} ] d[x] = ...

... (1/(n+1))·( f(x) )^{n+1} [o(x)o] ( f(x) )^{[o(x)o](-1)}

int[ ( ax+b )^{n} ] d[x] = ...

... (1/(n+1))·( ax+b )^{n+1} [o(x)o] (1/a)

int[ ( e^{ax+b} )^{n} ] d[x] = ...

... (1/(n+1))·( e^{ax+b} )^{n+1} [o(x)o] (-1)·e^{(-1)·(ax+b)} [o(x)o] (-1)·(1/a^{2})

int[ ( ln( ax+b ) )^{n} ] d[x] = ...

... (1/(n+1))·( ln( ax+b ) )^{n+1} [o(x)o] ln( ln( ax+b ) ) [o(x)o] (1/3)·( ax+b )^{3} [o(x)o] (1/a^{3})

int[ ( sin(ax+b) )^{n} ] d[x] = ...

... (1/(n+1))·( sin(ax+b) )^{n+1} [o(x)o] sin(ax+b) [o(x)o] (1/a^{2})+...

... (1/(n+1))·( sin(ax+b) )^{n+1} [o(x)o] tan[o(x)o](ax+b) [o(x)o] (-1)·cos(ax+b) [o(x)o] (1/a^{3})

int[ ( cos(ax+b) )^{n} ] d[x] = ...

... (1/(n+1))·( cos(ax+b) )^{n+1} [o(x)o] cos(ax+b) [o(x)o] (1/a^{2})+...

... (1/(n+1))·( cos(ax+b) )^{n+1} [o(x)o] cot[o(x)o](ax+b) [o(x)o] (-1)·sin(ax+b) [o(x)o] (1/a^{3})

int[ (-1)·( sin(ax+b) )^{n} ] d[x] = ...

... (1/(n+1))·( sin(ax+b) )^{n+1} [o(x)o] (-1)·sin(ax+b) [o(x)o] (1/a^{2})+...

... (1/(n+1))·( sin(ax+b) )^{n+1} [o(x)o] tan[o(x)o](ax+b) [o(x)o] cos(ax+b) [o(x)o] (1/a^{3})

int[ (-1)·( cos(ax+b) )^{n} ] d[x] = ...

... (1/(n+1))·( cos(ax+b) )^{n+1} [o(x)o] (-1)·cos(ax+b) [o(x)o] (1/a^{2})+...

... (1/(n+1))·( cos(ax+b) )^{n+1} [o(x)o] cot[o(x)o](ax+b) [o(x)o] sin(ax+b) [o(x)o] (1/a^{3})

int[ ( sinh(ax+b) )^{n} ] d[x] = ...

... (1/(n+1))·( sinh(ax+b) )^{n+1} [o(x)o] sinh(ax+b) [o(x)o] (1/a^{2})+...

... (1/(n+1))·( sinh(ax+b) )^{n+1} [o(x)o] tanh[o(x)o](ax+b) [o(x)o] (-1)·cosh(ax+b) [o(x)o] (1/a^{3})

int[ (-1)·( cosh(ax+b) )^{n} ] d[x] = ...

... (1/(n+1))·( cosh(ax+b) )^{n+1} [o(x)o] cosh(ax+b) [o(x)o] (1/a^{2})+...

... (1/(n+1))·( cosh(ax+b) )^{n+1} [o(x)o] coth[o(x)o](ax+b) [o(x)o] (-1)·sinh(ax+b) [o(x)o] (1/a^{3})

int[ (-1)·( sinh(ax+b) )^{n} ] d[x] = ...

... (1/(n+1))·( sinh(ax+b) )^{n+1} [o(x)o] (-1)·sinh(ax+b) [o(x)o] (1/a^{2})+...

... (1/(n+1))·( sinh(ax+b) )^{n+1} [o(x)o] tanh[o(x)o](ax+b) [o(x)o] cosh(ax+b) [o(x)o] (1/a^{3})

int[ ( cosh(ax+b) )^{n} ] d[x] = ...

... (1/(n+1))·( cosh(ax+b) )^{n+1} [o(x)o] (-1)·cosh(ax+b) [o(x)o] (1/a^{2})+...

... (1/(n+1))·( cosh(ax+b) )^{n+1} [o(x)o] coth[o(x)o](ax+b) [o(x)o] sinh(ax+b) [o(x)o] (1/a^{3})

int[ ( 1/sin(x) ) ] d[x] = (-1)·cos(x)+cot[o(x)o](x) [o(x)o] sin(x)

int[ (-1)·( 1/cos(x) ) ] d[x] = (-1)·sin(x)+tan[o(x)o](x) [o(x)o] cos(x)

int[ ( 1/cos(x) ) ] d[x] = sin(x)+tan[o(x)o](x) [o(x)o] (-1)·cos(x)

int[ (-1)·( 1/sin(x) ) ] d[x] = cos(x)+cot[o(x)o](x) [o(x)o] (-1)·sin(x)

cinematica: tren de (-b) a b

d_{t}[x] = (a/(2b))·( x^{2}+(-1)·b^{2} )

( ( 1/(x+(-b)) )+(-1)( 1/(x+b) ) )·d_{t}[x] = a

ln(x+(-b))+(-1)·ln(x+b) = at

1+(-1)·( (2b)/(x+b) ) = e^{at}

( (2b)/(x+b) ) = 1+(-1)·e^{at}

x(t) = ( (2b)/(1+(-1)·e^{at}) )+(-b)

d_{t}[x] = (2ba)·( e^{at}/(1+(-1)·e^{at})^{2} )

cinemática: tren de zero a b

d_{t}[x] = (a/b)·x(x+(-b))

( ( 1/(x+(-b)) )+(-1)·(1/x) )·d_{t}[x] = a

ln(x+(-b))+(-1)·ln(x) = at

(1+(-1)(b/x)) = e^{at}

(b/x)  = 1+(-1)·e^{at}

x(t) = ( b/(1+(-1)·e^{at}) )

d_{t}[x] = (ba)·( e^{at}/(1+(-1)·e^{at})^{2} )

Teorema del Buey del Projimo

( x € Local ) <==> No ( y € Local )

( f( objeto ) € x ) <==> No ( f( objeto ) € y )

( x € Sucursal-de-Caixa-Bank ) <==> No ( y € Sucursal-de-Caixa-Bank )

( f( Marihuana ) € x ) <==> No ( f( Marihuana ) € y )

sistemes de ecuacions diferencials

d_{t}[x(t)] = a·y(t)

d_{t}[x(t)] = b·z(t)

x(t) = (ab)·(1/(n+1))·t^{n+1}

y(t) = b·t^{n}

z(t) = a·t^{n}

d_{t}[x(t)] = a·e^{y(t)}

d_{t}[x(t)] = b·e^{z(t)}

x(t) = (ab)·(1/(n+1))·t^{n+1}

y(t) = ln(bt^{n})

z(t) = ln(at^{n})

d_{t}[x(t)] = a·ln(y(t))

d_{t}[x(t)] = b·ln(z(t))

x(t) = (ab)·(1/(n+1))·t^{n+1}

y(t) = e^{bt^{n}}

z(t) = e^{at^{n}}

d_{t}[x(t)] = a·( y(t) )^{p}

d_{t}[x(t)] = b·( z(t) )^{q}

x(t) = (ab)·(1/(n+1))·t^{n+1}

y(t) = b^{(1/p)}·t^{(n/p)}

z(t) = a^{(1/q)}·t^{(n/q)}

d_{t}[x(t)] = a·( y(t) )^{p}·f(y(t))

d_{t}[x(t)] = b·( z(t) )^{q}·f(z(t))

x(t) = (ab)·(1/(n+1))·t^{n+1}

y(t) = f-pow[p]( bt^{n} )

z(t) = f-pow[q]( at^{n} )

d_{t}[x(t)] = a·e^{u·y(t)}f(y(t))

d_{t}[x(t)] = b·e^{v·z(t)}·f(z(t))

x(t) = (ab)·(1/(n+1))·t^{n+1}

y(t) = f-e[u]( bt^{n} )

z(t) = f-e[v]( at^{n} )

d_{t}[x(t)] = a·( y(t) )^{p}·e^{u·y(t)}f(y(t))

d_{t}[x(t)] = b·( z(t) )^{q}·e^{v·z(t)}·f(z(t))

x(t) = (ab)·(1/(n+1))·t^{n+1}

y(t) = f-e[u]-pow[p]( bt^{n} )

z(t) = f-e[v]-pow[q]( at^{n} )

d_{t}[x(t)] = a·y(t)+(b/n)·t·d_{t}[z(t)]

d_{t}[x(t)] = b·z(t)+(a/n)·t·d_{t}[y(t)]

x(t) = (ab)·(1/(n+1))·t^{n+1}

y(t) = (b/2)·t^{n}

z(t) = (a/2)·t^{n}

d_{t}[x(t)] = a·e^{y(t)}+(b/n)·t·e^{z(t)}·d_{t}[z(t)]

d_{t}[x(t)] = b·e^{z(t)}+(a/n)·t·e^{y(t)}·d_{t}[y(t)]

x(t) = (ab)·(1/(n+1))·t^{n+1}

y(t) = ln((b/2)·t^{n})

z(t) = ln((a/2)·t^{n})

sistemes de ecuacions diferencials

sistema:

d_{t}[x(t)] = y(t)+z(t)

d_{t}[y(t)] = z(t)+x(t)

d_{t}[z(t)] = x(t)+y(t)

ecuación diferencial asociada al sistema:

d_{tt}^{2}[x(t)] = d_{t}[x(t)]+2·x(t)

d_{tt}^{2}[y(t)] = d_{t}[y(t)]+2·y(t)

d_{tt}^{2}[z(t)] = d_{t}[z(t)]+2·z(t)

k = (1/2)·( 1+3 ) = 2

x(t) = e^{2t}

y(t) = e^{2t}

z(t) = e^{2t}

sistema:

d_{t}[x(t)] = e^{y(t)}+e^{z(t)}

d_{t}[y(t)] = e^{z(t)}+e^{x(t)}

d_{t}[z(t)] = e^{x(t)}+e^{y(t)}

 

e^{2·x(t)} = e^{y(t)}e^{z(t)}

e^{2·y(t)} = e^{z(t)}e^{x(t)}

e^{2·z(t)} = e^{x(t)}e^{y(t)}

ecuación diferencial asociada a los sistemas:

d_{tt}^{2}[x(t)] = 2e^{2·x(t)}+e^{x(t)}·d_{t}[x(t)]

d_{tt}^{2}[y(t)] = 2e^{2·y(t)}+e^{y(t)}·d_{t}[y(t)]

d_{tt}^{2}[z(t)] = 2e^{2·z(t)}+e^{z(t)}·d_{t}[z(t)]

x(t) = ln(1/((-2)·t))

y(t) = ln(1/((-2)·t))

z(t) = ln(1/((-2)·t))

sistema:

d_{t}[x(t)] = ln(y(t))·y(t)+ln(z(t))·z(t)

d_{t}[y(t)] = ln(z(t))·z(t)+ln(x(t))·x(t)

d_{t}[z(t)] = ln(x(t))·x(t)+ln(y(t))·y(t)

2·( ln(x(t)) )^{2}·x(t) = ln(y(t))·ln(z(t))·( y(t)+z(t) )

2·( ln(y(t)) )^{2}·y(t) = ln(z(t))·ln(x(t))·( z(t)+x(t) )

2·( ln(z(t)) )^{2}·z(t) = ln(x(t))·ln(y(t))·( x(t)+y(t) )

2·( ln(x(t)) )^{2}·x(t) = ln(x(t))·x(t)·( ln(y(t))+ln(z(t)) )

2·( ln(y(t)) )^{2}·y(t) = ln(y(t))·y(t)·( ln(z(t))+ln(x(t)) )

2·( ln(z(t)) )^{2}·z(t) = ln(z(t))·z(t)·( ln(x(t))+ln(y(t)) )

ecuación diferencial asociada a los sistemas:

d_{tt}^{2}[x(t)] = d_{t}[x(t)]+2·ln(x(t))·x(t)+4·( ln(x(t)) )^{2}·x(t)

d_{tt}^{2}[y(t)] = d_{t}[y(t)]+2·ln(y(t))·y(t)+4·( ln(y(t)) )^{2}·y(t)

d_{tt}^{2}[z(t)] = d_{t}[z(t)]+2·ln(z(t))·z(t)+4·( ln(z(t)) )^{2}·z(t)

x(t) = e^{e^{2t}}

y(t) = e^{e^{2t}}

z(t) = e^{e^{2t}}

ecuacions diferencials sinus-exponencial-potencial

d_{x}[ e[n]-sin-arc[m](x) ] = d_{x}[ x^{m}e^{nx}sin(x) ] = ...

... (1/x)·( m·e[n]-sin-arc[m](x)+nx·e[n]-sin-arc[m](x)+...

... x·( x^{2m}e^{2nx}+(-1)·( e[n]-sin-arc[m](x) )^{2} )^{(1/2)} )

d_{y}[ arc[m]-sin-e[n](y) ]  = ...

... arc[m]-sin-e[n](y)·( 1/( m·y+ny·arc[m]-sin-e[n](y)+...

... arc[m]-sin-e[n](y)·( ( arc[m]-sin-e[n](y) )^{2m}e^{2n·arc[m]-sin-e[n](y)}+(-1)·y^{2} )^{(1/2)} ) )

f(x) = (1/y)·( (m+ny)·e[n]-sin-arc[m](y)+y·e[n]-cos-arc[m](y) )·d_{x}[y]

int[ f(x) ] d[x] = y^{m}·e^{nx}·sin(y)

int[ f(x) ] d[x] = e[n]-sin-arc[m](y)

arc[m]-sin-e[n]( int[ f(x) ] d[x] ) = y(x)