Homework — Dirac Delta FunctionsEvaluate the following1) f(3x2 +1)6(x)dx -.02) f(3x...

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Homework — Dirac Delta FunctionsEvaluate the following1) f(3x2 +1)6(x)dx -.02) f(3x XXXXXXXXXX5x)dx3) f(3x XXXXXXXXXX5x — 3)dx XXXXXXXXXXJ(3k2 +1)6(k — k')dk 5) The step function 0(x) is defined by0(x) = 0 x  0Is the derivative of the step function a Dirac delta function? Explain your answer with a calculation discussing1) Is the function 0 everywhere except one point? 2) Is the function really large at that one point? 3) Is the integral of the function in a reasonable interval equal to 1.

David · Accepted Answer

We wish to evaluate the following integrals involving the Dirac delta function.
1)    23 1 .I x x dx


 
In general, we have
       0 .f x x dx f



Substituting 23 1x   for  f x , we obtain 
 
  
 
2
3 0 1
1.
I  


2)    23 1 5 .I x x dx


 
Here we must first make a substitution to get the integral into the right form. We let
 5 ,u x
whence
2
2
2
3 1 3 1
5
3
1
25
u
x
u
 
   
 
 

and
 
1
.
5
dx du
Thus we have
 
2
2
2
3 1
1
25 5
1 3
1
5 25
3 01
1
5 25
1
.
5
u
I du
u
du




 
  
 
 
  
 
 
  
  




3)    23 1 5 3 .I x x dx


  
Once again, we solve by substitution. Letting
 5 3,u x 
we have
 
3
,

Homework — Dirac Delta Functions Evaluate the following 1) f(3x2 +1)6(x)dx -.0 2) f(3x XXXXXXXXXX5x)dx 3) f(3x XXXXXXXXXX5x — 3)dx XXXXXXXXXXJ(3k2 +1)6(k — k')dk 5) The step function 0(x) is defined...

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