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

Calculus Level 5

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

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

Then find the value of 02T(x)dx\int_{0}^{2}T(x)dx (correct upto 3 places after the decimal point)

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