integrals

int[ ln(x) ] d[x] = int[ y·e^{y} ] d[y] = y^{2}·[er-h]_{k!:2}(y) = ...

... ( ln(x) )^{2}·[er-h]_{k!:2}( ln(x) )

d_{x}[ ( ln(x) )^{2}·[er-h]_{k!:2}( ln(x) ) ] = ...

... ln(x)·(e^{ln(x)}/x) = ln(x)

int[ ln(1/x) ] d[x] = int[ (-1)·y·e^{(-y)} ] d[y] = (-1)·y^{2}·[er]_{k!:2}(y) = ...

... (-1)·( ln(1/x) )^{2}·[er]_{k!:2}( ln(1/x) )

d_{x}[ (-1)·( ln(1/x) )^{2}·[er]_{k!:2}( ln(1/x) ) ] = ...

... (-1)·ln(1/x)·( (-1)/x )·e^{(-1)·ln(1/x)} = ln(1/x)

int[ (1/ln(x)) ] d[x] = int[ (e^{y}/y) ] d[y] = [er-h]_{k!:0}(y) = ...

... [er-h]_{k!:0}( ln(x) )

d_{x}[ [er-h]_{k!:0}( ln(x) ) ] = ...

... (1/ln(x))·(e^{ln(x)}/x) = (1/ln(x))

int[ (1/ln(1/x)) ] d[x] = int[ (-1)·(e^{(-y)}/y) ] d[y] = (-1)·[er]_{k!:0}(y) = ...

... (-1)·[er]_{k!:0}( ln(1/x) )

d_{x}[ (-1)·[er]_{k!:0}( ln(1/x) ) ] = ...

... (-1)·(1/ln(1/x))·( (-1)/x )·e^{(-1)·ln(1/x)} = (1/ln(1/x))

int[ ln(ax+b) ] d[x] = ...

... ( ln(ax+b) )^{2}·[er-h]_{k!:2}( ln(ax+b) ) [o(x)o] (1/a)·x

int[ ln(1/(ax+b)) ] d[x] = ...

... (-1)·( ln(1/(ax+b)) )^{2}·[er]_{k!:2}( ln(1/(ax+b)) ) [o(x)o] (1/a)·x

int[ ln(ax^{2}+bx+c) ] d[x] = ...

... ( ln(ax^{2}+bx+c) )^{2}·[er-h]_{k!:2}( ln(ax^{2}+bx+c) ) [o(x)o] ...

... ln(2ax+b) [o(x)o] (1/(2a))·x

int[ ln(1/(ax^{2}+bx+c)) ] d[x] = ...

... (-1)·( ln(1/(ax^{2}+bx+c)) )^{2}·[er]_{k!:2}( ln(1/(ax^{2}+bx+c)) ) [o(x)o] ...

... ln(2ax+b) [o(x)o] (1/(2a))·x

int[ arc-sin(x) ] d[x] = int[ y·cos(y) ] d[y] = ...

... y^{2}·[er-cos]_{(2k)!:2}(y) = ...

... ( arc-sin(x) )^{2}·[er-cos]_{(2k)!:2}( arc-sin(x) )

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

cos(arc-sin(x)) = (1+(-1)·x^{2})^{(1/2)}

d_{x}[ ( arc-sin(x) )^{2}·[er-cos]_{(2k)!:2}( arc-sin(x) ) ] = ...

... arc-sin(x)·cos(arc-sin(x))·( 1/(1+(-1)·x^{2})^{(1/2)} ) = arc-sin(x)

int[ arc-cos(x) ] d[x] = int[ (-1)·y·sin(y) ] d[y] = ...

... (-1)·y^{2}·[er-sin]_{(2k+1)!:2}(y) = ...

... (-1)·( arc-cos(x) )^{2}·[er-sin]_{(2k+1)!:2}( arc-cos(x) )

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

sin(arc-cos(x)) = (1+(-1)·x^{2})^{(1/2)}

d_{x}[ ( arc-cos(x) )^{2}·[er-sin]_{(2k+1)!:2}( arc-cos(x) ) ] = ...

... (-1)·arc-cos(x)·sin(arc-cos(x))·( (-1)/(1+(-1)·x^{2})^{(1/2)} ) = arc-cos(x)

int[ x^{p}·arc-sin(x) ] d[x] = ...

... (1/(p+1))·x^{p+1} [o(x)o] ( arc-sin(x) )^{2}·[er-cos]_{(2k)!:2}( arc-sin(x) )

int[ x^{p}·arc-cos(x) ] d[x] = ...

... (1/(p+1))·x^{p+1} [o(x)o] (-1)·( arc-cos(x) )^{2}·[er-sin]_{(2k+1)!:2}( arc-cos(x) )

int[ arc-sinh(x) ] d[x] = int[ y·cosh(y) ] d[y] = ...

... y^{2}·[er-cosh]_{(2k)!:2}(y) = ...

... ( arc-sinh(x) )^{2}·[er-cosh]_{(2k)!:2}( arc-sinh(x) )

d_{x}[ cosh(arc-sinh(x)) ] = ( x/(1+x^{2})^{(1/2)} ) = d_{x}[ (1+x^{2})^{(1/2)} ]

cosh(arc-sinh(x)) = (1+x^{2})^{(1/2)}

d_{x}[ ( arc-sinh(x) )^{2}·[er-cosh]_{(2k)!:2}( arc-sinh(x) ) ] = ...

... arc-sinh(x)·cosh(arc-sinh(x))·( 1/(1+x^{2})^{(1/2)} ) = arc-sinh(x)

int[ arc-cosh(x) ] d[x] = int[ y·sinh(y) ] d[y] = ...

... y^{2}·[er-sinh]_{(2k+1)!:2}(y) = ...

... ( arc-cosh(x) )^{2}·[er-sinh]_{(2k+1)!:2}( arc-cosh(x) )

d_{x}[ sinh(arc-cosh(x)) ] = ( x/((-1)+x^{2})^{(1/2)} ) = d_{x}[ ((-1)+x^{2})^{(1/2)} ]

sinh(arc-cosh(x)) = ((-1)+x^{2})^{(1/2)}

d_{x}[ ( arc-cosh(x) )^{2}·[er-sinh]_{(2k+1)!:2}( arc-cosh(x) ) ] = ...

... arc-cosh(x)·sinh(arc-cosh(x))·( 1/((-1)+x^{2})^{(1/2)} ) = arc-cosh(x)

int[ tan(x) ] d[x] = int[ y/(1+(-1)·y^{2}) ]d[y] = (-1)·ln( (1+(-1)·y^{2})^{(1/2)} ) = ...

... (-1)·ln(cos(x))

int[ cot(x) ] d[x] = int[ (-y)/(1+(-1)·y^{2}) ]d[y] = ln( (1+(-1)·y^{2})^{(1/2)} ) = ...

... ln(sin(x))