So...It's not the gamma function?

Calculus Level 5

Let $I(n,a)=\int_{0}^{a}x^ne^{-x}dx$

Also define, $T(a)=\lim_{n\rightarrow \infty}(1-\frac{I(n,a)}{n!})e^a$

Then find the value of $\int_{0}^{2}T(x)dx$ (correct upto 3 places after the decimal point)

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