Small change in constants, big change in integral

[Rose Garden]

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?

 

[ForMyThunder]

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.

 

[Rose Garden]

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]

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.