Integrating 1/(Z^2 + A^2) - something wrong?

  • Thread starter Froskoy
  • Start date
  • #1
27
0

Homework Statement


[tex]\int \frac{1}{2t^2+4} \mathrm{d}t[\tex]


Homework Equations


[tex]\int \frac{1}{Z^2+A^2} \mathrm{d}Z = \frac{1}{A} \arctan{(\frac{Z}{A})} + c[/tex]


The Attempt at a Solution


Looks quite easy, but this is what's annoying me: the two methods below should be identical, but something's gone wrong and I can't work out what. For some reason, the second method gives an extra factor of 1/sqrt(2) at the front. Why aren't they both the same?

[tex]\int \frac{1}{2t^2+4} \mathrm{d}t = \int \frac{1}{(\sqrt{2t})^2+2^2} \mathrm{d}t = \frac{1}{2} \arctan {(\frac{\sqrt{2}t}{2})} + c[/tex]

[tex]\int \frac{1}{2t^2+4} \mathrm{d}t = \frac{1}{2} \int \frac{1}{t^2+2} = \frac{1}{2 \sqrt{2}} \arctan{(\frac{\sqrt{2} t}{2})}+c[/tex]

With very many thanks,

Froskoy.
 

Answers and Replies

  • #2
85
0
Are you sure that c represents the same value in each?
 
  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

Homework Statement


[tex]\int \frac{1}{2t^2+4} \mathrm{d}t[\tex]


Homework Equations


[tex]\int \frac{1}{Z^2+A^2} \mathrm{d}Z = \frac{1}{A} \arctan{(\frac{Z}{A})} + c[/tex]


The Attempt at a Solution


Looks quite easy, but this is what's annoying me: the two methods below should be identical, but something's gone wrong and I can't work out what. For some reason, the second method gives an extra factor of 1/sqrt(2) at the front. Why aren't they both the same?

[tex]\int \frac{1}{2t^2+4} \mathrm{d}t = \int \frac{1}{(\sqrt{2t})^2+2^2} \mathrm{d}t = \frac{1}{2} \arctan {(\frac{\sqrt{2}t}{2})} + c[/tex]

[tex]\int \frac{1}{2t^2+4} \mathrm{d}t = \frac{1}{2} \int \frac{1}{t^2+2} = \frac{1}{2 \sqrt{2}} \arctan{(\frac{\sqrt{2} t}{2})}+c[/tex]

With very many thanks,

Froskoy.

You forgot to use [itex] \sqrt{2} dt[/itex] in the first integral.

RGV
 

Related Threads on Integrating 1/(Z^2 + A^2) - something wrong?

  • Last Post
Replies
1
Views
4K
  • Last Post
Replies
5
Views
10K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
5
Views
732
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
7
Views
3K
Top