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...

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
,
5
u
x


whence
2
2 33 1...