Integral of $b[x]^{l[x]} = p[x]$

First year calculus student here - I am not sure where to turn for an answer despite searching and playing with Wolfram Alpha and Symbolab, and (forgive me) I do not know the technical vocabulary for what I am trying to do.

What is the equation which when differentiated with respect to x yields $b[x]^{l[x]} + C = p[x]$ ? Note these names were chosen based upon the relationship $base^{logarithm} = power$ (where, to be clear, "power" is NOT the exponent: i.e. $$b^l=p$$ $$\equiv log_b[p] = l$$ $$\equiv \sqrt[l]{p} = b$$ .

Knowing the integrals of $a^x$ (constant a) and $x^n$ (constant n) has not proved helpful so far.

More generally, what would the process of finding this be called? Is it in some sense an "inverse" of implicit differentiation?

• – Shaun Sep 7 '17 at 10:51
• Thanks Shaun, didn't even know the name for it. All a part of learning the professional tools. – CorvinoDiNevarca Sep 7 '17 at 10:56

If I understand the question, then for a given $b(x),l(x)$ you are asking for a function $p(x)$ so that $p'(x)=b(x)^{l(x)}$
The answer is, we don't know such expressions in general, in fact, sometimes it is just "impossible" to find an expression for the integral of those functions, for example $e^{-x^2}$ (which is $b(x)^{l(x)}$ for $b(x)=e$ and $l(x)=-x^2$).
check out this question Is there really no way to integrate $e^{-x^2}$?