0
$\begingroup$

Possible Duplicate:
Why can't you integrate all power functions without a log function?

We know that $f(r,x)=\int_{1}^xt^r dt=\frac x{1+r}+C$ if $r\neq -1$ and $=\log x+C$ if $r=-1$. I find this quite strange. It's a weird singularity right there, where integrating a monomial $x^r$ would "escape" to another class of equations only at $r=-1$.

  1. Is there any explanation for this phenomenon?

  2. What about similar singular behaviors in other such functions?

  • 0
    See also the second explanation in my answer at http://math.stackexchange.com/questions/129849/linear-homogeneous-recurrence-relations-with-repeated-roots-motivation-behind-l/129855#129855 for another example of this phenomenon.2012-04-24

1 Answers 1

1

Here's a rather broad class of examples. Suppose $F(x)$ is a differentiable function with $F'(x) = f(x)$. Then $\int f(\lambda x)\ dx = \lambda^{-1} F(\lambda x) + C$ for $\lambda \ne 0$. Of course that doesn't make sense for $\lambda = 0$, where $\int f(0x)\ dx = f(0) x + C$ instead.

EDIT: At first sight your example of $\int t^r \ dt$ doesn't appear to fit this paradigm, but if you first do the change of variables $t = e^x$ it becomes $\int e^{(r+1) x}\ dx$ which does.