# Is there a general method to go about deriving a definite integral for a given result?

I was reading a blog post earlier about the Sophomore's Dream and a question came to mind:

Say we wanted to find a definite integral that gives the following result

$$\sum_{n=1}^\infty \left(\frac{a}{n}\right)^n=a+\left(\frac{a}{2}\right)^2+\left(\frac{a}{3}\right)^3+\cdots$$

My guess would be that we'd try to find a function $f(a)$ such that

$$\int^\beta_\alpha f(a)\,\mathrm{d}a = \sum_{n=1}^\infty \left(\frac{a}{n}\right)^n$$

assuming that it's indeed $f(a)$ we want (and not of some other variable)...

Is this an integral equation (I'm quite unfamiliar with them), and if so, is there a popular method to solve those of this form?

Or could we just use the FTC or something to solve for $f$?

Otherwise, how would you go about deriving such an indefinite integral?

• It doesn't make sense to have $a$ as a bound on the integral as well as being the variable which the integral is taken with respect to. – Will Fisher Jul 24 '16 at 1:17
• oops - you're right. fixing it. – galois Jul 24 '16 at 1:32