# Infinite Integrations?!?!

Calculus Level 5

Let the derivative of $$F(x)$$ be $$f(x)$$. Therefore, in general,
$\int f(x) dx = F(x) + c$
However, we define a new function $$\psi$$ that takes a function and returns the indefinite integral of the function with the constant as zero.Therefore,
$\psi (f(x)) = F(x)$

(Note that $$c = 0$$ )
( The whole idea is to make that sure that in any integration during the sum the constant is assumed to be zero. I am sorry for any technical mistake while explaining the idea.)

$$\psi^{n+1}(g(x)) = \psi(\psi^{n}(g(x)))$$ and $$\psi^{1}(g(x)) = \psi(g(x)) \forall n \in \mathbb{N}$$

Let $$\psi^{n}( \ln x )|_{ x =(1 + \frac{1}{n}) } = \phi(n)$$

If, $$L = \displaystyle \lim _ { n \to \infty} (n!)$$ $$\phi(n) + \ln (n^e)$$

Find the value of $$e + L$$

Details and Assumptions:

• $$\ln x$$ denotes the natural logarithm of $$x$$.

• $$n!$$ denotes the factorial of $$n$$.

• $$e$$ is the exponential constant.

×