Repeated Integral

Author

Eric W. Weisstein
April 15, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/RepeatedIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RepeatedIntegral.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Examples

testIntegrate[f_,x_,n_]:={Nest[Integrate[#,x]&,f,n],Integrate[(f/.x->t)(x-t)^(n-1)/(n-1)!,{t,0,x}],
Integrate[f,x]/.x->0}

testIntegrate[x^3,x,3]

Simplify[testIntegrate[Sin[x],x,2]]

Simplify[testIntegrate[Cos[x],x,1]]

Simplify[testIntegrate[x^2+6x,x,3]]

RepeatedIntegral[f_,{x_,n_}]:=Integrate[Function[x,f][t](x-t)^(n-1)/(n-1)!,{t,0,x}]

RepeatedIntegral[f_,{x_,n_}]:=Integrate[Function[x,f][t](x-t)^(n-1)/(n-1)!,t]/.t->x

Simplify[Table[RepeatedIntegral[Cos[x],{x,n}],{n,4}]]

Simplify[Table[RepeatedIntegral[Exp[x],{x,n}],{n,4}]]

Simplify[Table[Nest[Integrate[#,x]&,Exp[x],n],{n,5}]]