Integrate question
- To: mathgroup at smc.vnet.net
- Subject: [mg77600] Integrate question
- From: dimitris <dimmechan at yahoo.com>
- Date: Wed, 13 Jun 2007 07:44:47 -0400 (EDT)
This question aose after replying in the forum for another CAS.
In fact, it is not a question on "how I do something" but rather
asking about some information about "why".
In view of my reply I realized one fundamental difference
between Integrate and the function Int from another CAS.
int( diff(y(x),x) , x=a..b);
b
/
| d
| -- y(x) dx
| dx
/
a
In[11]:=
Integrate[Derivative[1][f][x], {x, a, b}]
Out[11]=
-f[a]+f[b]
I guess all we are familiar with the first fundamental theorem of
calculus.
It holds (http://mathworld.wolfram.com/
FundamentalTheoremsofCalculus.html)
if the function f[x] is continuous on the closed interval [a,b] (or at
least piecewise
continuous in Newton-Leibniz form with limits left and right of the
finite discontinuities).
I am with Mathematica default design. For example
In[12]:=
Integrate[Derivative[1][f][x]*Derivative[2][f][x], {x, a, b}]
Out[12]=
(1/2)*(-Derivative[1][f][a]^2 + Derivative[1][f][b]^2)
In[13]:=
Integrate[Derivative[1][f][x]*g[x] + f[x]*Derivative[1][g][x], {x, a,
b}]
Out[13]=
(-f[a])*g[a] + f[b]*g[b]
In[14]:=
Integrate[Sin[f[x]]*Derivative[1][f][x], {x, a, b}]
Out[14]=
Cos[f[a]] - Cos[f[b]]
I am neither software enginner, nor pure mathematician but this
fundmental difference impressed me a lot! I am familiar
with the issue of generic complex values in mathematica
but here Mathematica "assumes" that typing Integrate[f'[x],{x,a,b}]
f[x] is continuous in [a,b]?
I will appreciate any comments on this issue.
Greeting from the sunny Athens,
Dimitris
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