Re: Change integral variables

*To*: mathgroup at smc.vnet.net*Subject*: [mg88216] Re: Change integral variables*From*: samuel.blake at maths.monash.edu.au*Date*: Mon, 28 Apr 2008 04:41:23 -0400 (EDT)*References*: <fv1fep$es1$1@smc.vnet.net>

On Apr 27, 7:02 pm, Budaoy <yaomengli... at gmail.com> wrote: > In Mathematica, how can I change integral variables? For example in > integration: > > Integrate[Sin[Sqrt[x]], x] if I want to use t^2=x to instead x in the > integral, how can I achieve this? > > PS: Does Mathematica have a inertial form of some symbolic command, > for instance, the above integration, if I only want an integration > form but not an answer, what can I do? > > Thanks. Greetings, The following is a rough implementation of an integration by substitution routine.... In[87]:= IntegrateBySubstitution[integrand_, var_, sub_Equal] := Catch[Module[{u, substitution, du, res, x}, u = First[sub]; substitution = Last[sub]; du = D[substitution, var]; res = Simplify[(integrand /. substitution :> u)/du]; (* derivative cancels *) If[FreeQ[res, var], Throw[{res, u}] ]; (* use inverse function to remove var from res *) x = First@Flatten@Solve[sub, var]; res = PowerExpand@Simplify[res /. x]; {res, u} ]] Here is the integral you requested: In[88]:= IntegrateBySubstitution[Sin[Sqrt[x]], x, u == Sqrt[x]] Out[88]= {2 u Sin[u], u} And a couple of other examples ;-) In[89]:= IntegrateBySubstitution[(2 x)/(1 + x^2), x, u == 1 + x^2] Out[89]= {1/u, u} In[90]:= IntegrateBySubstitution[(2 x)/(1 + x^2), x, u == x^2] Out[90]= {1/(1 + u), u} In[91]:= IntegrateBySubstitution[Sqrt[1 - x + Sqrt[x]], x, u == Sqrt[x]] Out[91]= {2 u Sqrt[1 + u - u^2], u} In[92]:= IntegrateBySubstitution[Sqrt[1 + Tan[x]], x, u == Tan[x]] During evaluation of In[92]:= Solve::ifun: Inverse functions are \ being used by Solve, so some solutions may not be found; use Reduce \ for complete solution information. >> Out[92]= {Sqrt[1 + u]/(1 + u^2), u} Cheers, Sam