Small change in constants, big change in integral

  • #1

Main Question or Discussion Point

Check this out,
1/(x2+a)
where "a" is a constant

When this function is integrated, if a is positive then we get something like arctan of something, if a is 0 we simply get -1/x, and if a is negative then we get something involving the natural logarithm, and yet there's something very similar to all 3 graphs.

But how is it that a small change in this constant a can lead to such drastic changes in the functional form of the integral?
 

Answers and Replies

  • #2
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They are very similar. Generally, you factor the denominator, then use partial fractions. This is what you do when a is negative, or zero. When a is positive, you can do the same thing, but you get complex roots. Then you procede in the normal way and use the relation arctan(z)=(i/2)ln[(1-iz)/(1+iz)] and you should get the same answer.
 
  • #3
thanks, I have no problem integrating these, I'm just really amazed by how a change in some constant can lead to integrals of completely different forms, and yet still look very similar, is there any way to explain this?
 
  • #4
AlephZero
Science Advisor
Homework Helper
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The similarities are more obvious if you think about functions of complex variables.

[tex]e^{iz} = \cos z + i \sin z[/tex]

In a sense, "trig functions" and "exponentials" are only different names for the same basic mathematical function.

The same applies to their inverse functions, like log and arctan.

The derivative of 1/x is log x, so in that sense 1/x is also related to the others.
 

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