# Small change in constants, big change in integral

## 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?

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.

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?

AlephZero
Homework Helper
The similarities are more obvious if you think about functions of complex variables.

$$e^{iz} = \cos z + i \sin z$$

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.