SamplePublisher`GrassmannCalculus`
IntegralOperateRules  |
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In[2]:=
Inactive[Integrate][(f[x]+g[x])(f[x]-g[x]),x]%/.
[Expand]%//
 | IntegralOperateRules |
 | IntegralBreakout |
Out[2]=
∫(f[x]-g[x])(f[x]+g[x])x
Out[2]=
∫(-)x
2
f[x]
2
g[x]
Out[2]=
∫x-∫x
2
f[x]
2
g[x]
In[3]:=
Inactive[Integrate][(f[x]+g[x])(f[x]-g[x]),{x,a,b},{y,c,d}]%/.
[Expand]%//
| IntegralOperateRules |
| IntegralBreakout |
Out[3]=
b
∫
a
d
∫
c
Out[3]=
b
∫
a
d
∫
c
2
f[x]
2
g[x]
Out[3]=
b
∫
a
d
∫
c
2
f[x]
b
∫
a
d
∫
c
2
g[x]
Here we perform a trigonometric expansion of the integrand and separate into two integrals to see their individual contributions.
In[4]:=
SetEuclideanNSpace[2,{x,y},"Form"]
[Ball[{1,0}],Sin[x+y]Cos[x-y]dx⋀dy]%/.
[TrigExpand]List@@%//
%//
%//N
| FormIntegral |
| IntegralOperateRules |
 | IntegralBreakout |
| EvaluateFormIntegrals |
Out[4]=
∫
ℴ
Out[4]=
∫
ℴ
Out[4]=
Cos[x]Sin[x]dx⋀dy,Cos[y]Sin[y]dx⋀dy
∫
ℴ
∫
ℴ
Out[4]=
πHypergeometric0F1Regularized[2,-1]Sin[2],0
1
2
Out[4]=
{0.823748,0.}
Another use of the operate rule might be to perform partial fraction expansions.
In[5]:=
Inactive[Integrate],x%/.
[Apart]%//
//SimplifyActivate[%]+C
x+3
(x-a)(x-b)
| IntegralOperateRules |
| IntegralBreakout |
Out[5]=
x
3+x
(-a+x)(-b+x)
Out[5]=
+x
3+a
(a-b)(-a+x)
-3-b
(a-b)(-b+x)
Out[5]=
(3+a)x-(3+b)x
1
-a+x
1
-b+x
a-b
Out[5]=
C+
(3+a)Log[-a+x]-(3+b)Log[-b+x]
a-b
 |
|
 |
|
""