transformada integral

d_{xx}^{2}[y(x)]+y(x) = f(x)

d_{xx}^{2}[y(x)]·sin(x)+y(x)·sin(x) = f(x)·sin(x)

d_{xx}^{2}[y(x)]·cos(x)+y(x)·cos(x) = f(x)·cos(x)

d_{x}[ d_{x}[y(x)]·sin(x)+y(x)·(-1)·cos(x) ] = f(x)·sin(x)

d_{x}[ d_{x}[y(x)]·cos(x)+y(x)·sin(x) ] = f(x)·cos(x)

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

d_{x}[y(x)]·cos(x)+y(x)·sin(x) = int[ f(x)·cos(x) ] d[x]

d_{x}[y(x)]·( sin(x) )^{2}+y(x)·(-1)·sin(x)·cos(x) = sin(x)·int[ f(x)·sin(x) ] d[x]

d_{x}[y(x)]·( cos(x) )^{2}+y(x)·sin(x)·cos(x) = cos(x)·int[ f(x)·cos(x) ] d[x]

y(x) = int[ sin(x)·int[ f(x)·sin(x) ] d[x] ] d[x]+int[ cos(x)·int[ f(x)·cos(x) ] d[x] ] d[x]

cos(x) = (1/2)·( int[ e^{(p+i)x}] d[x]+int[ e^{(p+(-i))x}] d[x] ) = ( p/(p^{2}+1) )

sin(x) = (1/2i)·( int[ e^{(p+i)x}] d[x]+(-1)·int[ e^{(p+(-i))x}] d[x] ) = ( 1/(p^{2}+1) )

( cos(x) )^{2} = ...

... (1/4)·( int[ e^{(p+2i)x}] d[x]+int[ e^{(p+(-2)·i)x}] d[x] )+...

... (1/2)·int[ e^{px}] d[x] = (1/2)·( p/(p^{2}+4) )+( 1/(2p) )

( sin(x) )^{2} = ...

... (-1)·(1/4)·( int[ e^{(p+2i)x}] d[x]+int[ e^{(p+(-2)·i)x}] d[x] )+...

... (1/2)·int[ e^{px}] d[x] = (-1)·(1/2)·( p/(p^{2}+4) )+( 1/(2p) )

( (1/2)·p )+( 0+(1/2)·p ) = p

( (1/2)·p )+( 1+(-1)·(1/2)·p ) = 1

( sin(x) )^{[o(x)o]2} = ...

... (1/4)·(1/2i)·( int[ e^{(p+2i)x}] d[x]+(-1)·int[ e^{(p+(-2)·i)x}] d[x] )+...

... (1/2)·int[ x·e^{px}] d[x] = (-1)·(1/2)·( 1/(p^{2}+4) )+(-1)·( 1/(2p^{2}) )

( (-1)·cos(x) )^{[o(x)o]2} = ...

... (-1)·(1/4)·(1/2i)·( int[ e^{(p+2i)x}] d[x]+(-1)·int[ e^{(p+(-2)·i)x}] d[x] ) = ...

... (1/2)·int[ x·e^{px}] d[x] = (1/2)·( 1/(p^{2}+4) )+(-1)·( 1/(2p^{2}) )

( sin(x) )^{[o( (1/(2k)!)·x^{(2k)} )o]2} = ...

... (-1)·(1/2)·(1/4^{k})·( p/(p^{2}+4) )+(-1)^{(2k)}·( 1/(2p^{(2k+1)}) )

( cos(x) )^{[o( (1/(2k)!)·x^{(2k)} )o]2} = ...

... (1/2)·(1/4^{k})·( p/(p^{2}+4) )+(-1)^{(2k)}·( 1/(2p^{(2k+1)}) )

( sin(x) )^{[o( (1/(2k+1)!)·x^{(2k+1)} )o]2} = ...

... (-1)·(1/2)·(1/4^{k})·( 1/(p^{2}+4) )+(-1)^{(2k+1)}·( 1/(2p^{(2k+2)}) )

( cos(x) )^{[o( (1/(2k+1)!)·x^{(2k+1)} )o]2} = ...

... (1/2)·(1/4^{k})·( 1/(p^{2}+4) )+(-1)^{(2k+1)}·( 1/(2p^{(2k+2)}) )

d_{x}[y(x)]+py(x) = ( cos(x) )

d_{x}[z(x)]+pz(x) = ( sin(x) )

d_{x}[y(x)]·e^{px}+py(x)·e^{px} = ( cos(x) )·e^{px}

d_{x}[z(x)]·e^{px}+pz(x)·e^{px} = ( sin(x) )·e^{px}

d_{x}[ y(x)·e^{px} ] = ( cos(x) )·e^{px}

d_{x}[ z(x)·e^{px} ] = ( sin(x) )·e^{px}

y(x) = ( sin(x) [o(x)o] (1/p)·e^{px} )·e^{(-p)x}

z(x) = ( (-1)·cos(x) [o(x)o] (1/p)·e^{px} )·e^{(-p)x}

Tra[y(x)] = ( 1/(p^{2}+1) [o(+)o] (1/p^{2}) )+(1/(-p))

Tra[z(x)] = ( (-p)/(p^{2}+1) [o(+)o] (1/p^{2}) )+(1/(-p))

d_{x}[ Tra[y(x)] ] = ( p/(p^{2}+1) [+] ( 1/(p^{2}+1) [o(+)o] (1/p^{2}) )+1

d_{x}[ Tra[z(x)] ] = ( 1/(p^{2}+1) [+] ( (-p)/(p^{2}+1) [o(+)o] (1/p^{2}) )+1

p·Tra[y(x)] = ( 1/(p^{2}+1) [o(+)o] (1/p^{2}) )+(-1)

p·Tra[z(x)] = ( (-p)/(p^{2}+1) [o(+)o] (1/p^{2}) )+(-1)

d_{x}[ Tra[y(x)] ] [+] p·Tra[y(x)] = ( p/(p^{2}+1)

d_{x}[ Tra[z(x)] ] [+] p·Tra[z(x)] = ( 1/(p^{2}+1)

d_{x}[ ( 1/(p+s) ) ] = ( 1/(p+s) )·(p+s)

int[ ( 1/(p+s) ) ] d[x] = ( 1/(p+s) )·( 1/(p+s) )

d_{x}[ ( 1/(-p) ) ] = ( 1/(-p) )·(-p)

int[ ( 1/(-p) ) ] d[x] = ( 1/(-p) )·( 1/(-p) )

d_{x}[ sin(x) ] = cos(x)

d_{x}[ cos(x) ] = (-1)·sin(x)

d_{x}[ (-1)·sin(x) ] = (-1)·cos(x)

d_{x}[ (-1)·cos(x) ] = sin(x)

d_{x}[ ( 1/(p^{2}+1) ) ] = ( 1/(p^{2}+1) )·p

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

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

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

int[ ( 1/(p^{2}+1) ) ] d[x] = ( 1/(p^{2}+1) )·(-p)

int[ ( p/(p^{2}+1) ) ] d[x] = ( p/(p^{2}+1) )·( 1/p )

int[ ( (-1)/(p^{2}+1) ) ] d[x]= ( (-1)/(p^{2}+1) )·(-p)

int[ ( (-p)/(p^{2}+1) ) ] d[x]= ( (-p)/(p^{2}+1) )·( 1/p )

d_{x}[ sin(ax) ] = cos(ax)·a

d_{x}[ cos(ax) ] = (-1)·sin(ax)·a

d_{x}[ (-1)·sin(ax) ] = (-1)·cos(ax)·a

d_{x}[ (-1)·cos(ax) ] = sin(ax)·a

d_{x}[ ( a/(p^{2}+a^{2}) ) ] = ( a/(p^{2}+a^{2}) )·p

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

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

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

(p+4)/(p^{2}+8p+41) = (p+4)/((p+4)^{2}+25) = e^{4x}·cos(5x)

(p+2)/(p^{2}+8p+41) = ((p+4)+(-2))/((p+4)^{2}+25) = e^{4x}·cos(5x)+(-1)·(2/5)·sin(5x)

d_{x}[y(x)]+py(x) = ( e^{ax}·cos(bx) )

d_{x}[z(x)]+pz(x) = ( e^{ax}·sin(bx) )

d_{x}[y(x)]·e^{px}+py(x)·e^{px} = ( cos(bx) )·e^{(p+a)x}

d_{x}[z(x)]·e^{px}+pz(x)·e^{px} = ( sin(bx) )·e^{(p+a)x}

d_{x}[ y(x)·e^{px} ] = ( cos(bx) )·e^{(p+a)x}

d_{x}[ z(x)·e^{px} ] = ( sin(bx) )·e^{(p+a)x}

y(x) = ( (1/b)·sin(bx) [o(x)o] (1/(p+a))·e^{(p+a)x} )·e^{(-p)x}

z(x) = ( (-1)·(1/b)·cos(bx) [o(x)o] (1/(p+a))·e^{(p+a)x} )·e^{(-p)x}

Tra[y(x)] = ( (1/b)·( b/(p^{2}+b^{2}) ) [o(+)o] (1/(p+a)^{2}) )+( 1/(-p) )

Tra[z(x)] = ( (1/b)·( (-p)/(p^{2}+b^{2}) ) [o(+)o] (1/(p+a)^{2}) )+( 1/(-p) )

Tra[y(x)] = ( (1/b)·( b/((p+a)^{2}+b^{2}) ) [o(+)o] (1/p^{2}) )+(1/(-p))

Tra[z(x)] = ( (1/b)·( (-1)·(p+a)/((p+a)^{2}+b^{2}) ) [o(+)o,,] (1/p^{2}) )+(1/(-p))