How-can-i-evaluate-the-value-of-2-4-e-t-t-dt-

Question Number 150064 by n0y0n last updated on 09/Aug/21

$$\mathrm{How}\:\mathrm{can}\:\mathrm{i}\:\mathrm{evaluate}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\: \\ $$$$\:\:\int_{\mathrm{2}} ^{\:\mathrm{4}} \frac{\mathrm{e}^{\mathrm{t}} }{\mathrm{t}}\mathrm{dt}\:=\:? \\ $$

Answered by Ar Brandon last updated on 09/Aug/21

∫_2 ^4 Σ_(n=0) ^∞ (t^(n−1) /(n!))dt=ln2+Σ_(n≥1) ((4^n −2^n )/(n(n!)))+C

$$\int_{\mathrm{2}} ^{\mathrm{4}} \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{t}^{{n}−\mathrm{1}} }{{n}!}{dt}=\mathrm{ln2}+\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{4}^{{n}} −\mathrm{2}^{{n}} }{{n}\left({n}!\right)}+{C} \\ $$

Commented by n0y0n last updated on 09/Aug/21